শর্তমতে,
\(u=u(x)\) এবং \(w=w(x)\)
ধরি,
\(v=\frac{d}{dx}(w)\)
\(\Rightarrow \int{vdx}=\int{\frac{d}{dx}(w)dx}\) ➜Note উভয় পার্শে \(x\)-এর সাপেক্ষে যোগজীকরণ করে।
\(\Rightarrow \int{vdx}=\int{dw}\)
\(\Rightarrow \int{vdx}=w\) ➜Note যোগজীকরণ ধ্রুবক \((c)\) উপেক্ষা করে ।
\(\Rightarrow u\int{vdx}=uw\) ➜Note উভয় পার্শে \(u\) গুণ করে ।
\(\Rightarrow \frac{d}{dx}\left(u\int{vdx}\right)=\frac{d}{dx}(uw)\) ➜Note উভয় পার্শে \(x\)-এর সাপেক্ষে অন্তরীকরণ করে।
\(\Rightarrow \frac{d}{dx}\left(u\int{vdx}\right)=u\frac{d}{dx}(w)+w\frac{d}{dx}(u)\) ➜Note \(\because \frac{d}{dx}(uv)=u\frac{d}{dx}(v)+v\frac{d}{dx}(u)\)
\(\Rightarrow \frac{d}{dx}\left(u\int{vdx}\right)=uv+\int{vdx}.\frac{d}{dx}(u)\) ➜Note \(\because \frac{d}{dx}(w)=v, w=\int{vdx}\)
\(\Rightarrow \frac{d}{dx}\left(u\int{vdx}\right)=uv+\frac{d}{dx}(u)\int{vdx}\)
\(\Rightarrow \int{\frac{d}{dx}\left(u\int{vdx}\right)dx}=\int{(uv)dx}+\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\) ➜Note উভয় পার্শে \(x\)-এর সাপেক্ষে যোগজীকরণ করে।
\(\Rightarrow \int{d\left(u\int{vdx}\right)}=\int{(uv)dx}+\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\) \(\Rightarrow u\int{vdx}=\int{(uv)dx}+\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\) ➜Note \(\because \int{dx}=x\)
\(\Rightarrow \int{(uv)dx}+\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}=u\int{vdx}\)
\(\therefore \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)

\(\int{\ln{|x|}dx}\)
\(=\int{\ln{|x|}.1dx}\)
\(=\ln{|x|}\int{1dx}-\int{\left\{\frac{d}{dx}(\ln{|x|})\int{1dx}\right\}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\ln{|x|}.x-\int{\left\{\frac{1}{x}.x\right\}dx}\) ➜Note \(\because \int{1dx}=x, \frac{d}{dx}(\ln{|x|})=\frac{1}{x}\)
\(=x\ln{|x|}-\int{1dx}\)
\(=x\ln{|x|}-x+c\) ➜Note \(\because \int{1dx}=x\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=x(\ln{|x|}-1)+c\)

\(\int{\tan^{-1}{x}dx}\)
\(=\int{\tan^{-1}{x}.1dx}\)
\(=\tan^{-1}{x}\int{1dx}-\int{\left\{\frac{d}{dx}(\tan^{-1}{x})\int{1dx}\right\}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\tan^{-1}{x}.x-\int{\left\{\frac{1}{1+x^2}.x\right\}dx}\) ➜Note \(\because \int{1dx}=x, \frac{d}{dx}(\tan^{-1}{x})=\frac{1}{1+x^2}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=x\tan^{-1}{x}-\int{\frac{1}{1+x^2}xdx}\)
\(=x\tan^{-1}{x}-\int{\frac{1}{t}\times{\frac{1}{2}}dt}\)
\(=x\tan^{-1}{x}-\frac{1}{2}\int{\frac{1}{t}dt}\)
\(=x\tan^{-1}{x}-\frac{1}{2}\ln{\left|t\right|}+c\) ➜Note \(\because \int{\frac{1}{x}dx}=\ln{|x|}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=x\tan^{-1}{x}-\frac{1}{2}\ln{\left|1+x^2\right|}+c\)

ধরি,
\(I=\int{e^x\sin{x}dx}\)
\(\Rightarrow I=\int{\sin{x}.e^xdx}\)
\(\Rightarrow I=\sin{x}\int{e^xdx}-\int{\left\{\frac{d}{dx}(\sin{x})\int{e^xdx}\right\}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(\Rightarrow I=\sin{x}.e^x-\int{\left\{\cos{x}.e^x\right\}dx}\) ➜Note \(\because \int{e^xdx}=e^x, \frac{d}{dx}(\sin{x})=\cos{x}\)
\(\Rightarrow I=e^x\sin{x}-\int{\cos{x}.e^xdx}\)
\(\Rightarrow I=e^x\sin{x}-\left[\cos{x}\int{e^xdx}-\int{\left\{\frac{d}{dx}(\cos{x})\int{e^xdx}\right\}dx}\right]\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(\Rightarrow I=e^x\sin{x}-\left[\cos{x}.e^x-\int{(-\sin{x}).e^xdx}\right]\) ➜Note \(\because \int{e^xdx}=e^x, \frac{d}{dx}(\cos{x})=-\sin{x}\)
\(\Rightarrow I=e^x\sin{x}-\left[e^x\cos{x}+\int{e^x\sin{x})dx}\right]\)
\(\Rightarrow I=e^x\sin{x}-\left[e^x\cos{x}+I\right]\) ➜Note \(\because \int{e^x\sin{x}dx}=I\)
\(\Rightarrow I=e^x\sin{x}-e^x\cos{x}-I\)
\(\Rightarrow I+I=e^x\sin{x}-e^x\cos{x}\)
\(\Rightarrow 2I=e^x\sin{x}-e^x\cos{x}\)
\(\Rightarrow I=\frac{1}{2}(e^x\sin{x}-e^x\cos{x})+c\) ➜Note\(c\) যোগজীকরণ ধ্রুবক।
\(\therefore \int{e^x\sin{x}dx}=\frac{1}{2}(e^x\sin{x}-e^x\cos{x})+c\)

\(L.S=I=\int{e^{ax}\sin{bx}dx}\)
\(\Rightarrow I=\int{\sin{bx}.e^{ax}dx}\)
\(\Rightarrow I=\sin{bx}\int{e^{ax}dx}-\int{\left\{\frac{d}{dx}(\sin{bx})\int{e^{ax}dx}\right\}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(\Rightarrow I=\sin{bx}.\frac{e^{ax}}{a}-\int{\left\{\cos{bx}.\frac{d}{dx}(bx).\frac{e^{ax}}{a}\right\}dx}\) ➜Note \(\because \int{e^{ax}dx}=\frac{e^{ax}}{a}, \frac{d}{dx}(\sin{x})=\cos{x}\)
\(\Rightarrow I=\frac{1}{a}e^{ax}\sin{bx}-\int{\left\{\cos{bx}.b.\frac{e^{ax}}{a}\right\}dx}\)
\(\Rightarrow I=\frac{1}{a}e^{ax}\sin{bx}-\frac{b}{a}\int{\cos{bx}.e^{ax}dx}\)
\(\Rightarrow I=\frac{1}{a}e^{ax}\sin{bx}-\frac{b}{a}\left[\cos{bx}\int{e^{ax}dx}-\int{\left\{\frac{d}{dx}(\cos{bx})\int{e^{ax}dx}\right\}dx}\right]\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(\Rightarrow I=\frac{1}{a}e^{ax}\sin{bx}-\frac{b}{a}\left[\cos{bx}.\frac{e^{ax}}{a}-\int{(-\sin{bx}).\frac{d}{dx}(bx).\frac{e^{ax}}{a}dx}\right]\) ➜Note \(\because \int{e^{ax}dx}=\frac{e^{ax}}{a}, \frac{d}{dx}(\cos{x})=-\sin{x}\)
\(\Rightarrow I=\frac{1}{a}e^{ax}\sin{bx}-\frac{b}{a}\left[\frac{1}{a}e^{ax}\cos{bx}+\int{\sin{bx}.b.\frac{e^{ax}}{a}dx}\right]\)
\(\Rightarrow I=\frac{1}{a}e^{ax}\sin{bx}-\frac{b}{a}\left[\frac{1}{a}e^{ax}\cos{bx}+\frac{b}{a}\int{\sin{bx}.e^{ax}dx}\right]\)
\(\Rightarrow I=\frac{1}{a}e^{ax}\sin{bx}-\frac{b}{a^2}e^{ax}\cos{bx}-\frac{b^2}{a^2}\int{\sin{bx}.e^{ax}dx}\)
\(\Rightarrow I=\frac{1}{a}e^{ax}\sin{bx}-\frac{b}{a^2}e^{ax}\cos{bx}-\frac{b^2}{a^2}I\) ➜Note \(\because \int{\sin{bx}.e^{ax}dx}=I\)
\(\Rightarrow I+\frac{b^2}{a^2}I=\frac{1}{a}e^{ax}\sin{bx}-\frac{b}{a^2}e^{ax}\cos{bx}\)
\(\Rightarrow I\left(1+\frac{b^2}{a^2}\right)=\frac{1}{a^2}e^{ax}(a\sin{bx}-b\cos{bx})\)
\(\Rightarrow I\left(\frac{a^2+b^2}{a^2}\right)=\frac{1}{a^2}e^{ax}(a\sin{bx}-b\cos{bx})\)
\(\Rightarrow I=\frac{a^2}{a^2+b^2}\times{\frac{1}{a^2}}e^{ax}(a\sin{bx}-b\cos{bx})+c\) ➜Note\(c\) যোগজীকরণ ধ্রুবক।
\(\Rightarrow I=\frac{e^{ax}}{a^2+b^2}(a\sin{bx}-b\cos{bx})+c\)
\(\Rightarrow I=R.S\)
\(\therefore L.S=R.S\)
(Proved)

\(L.S=I=\int{e^{ax}\cos{bx}dx}\)
\(\Rightarrow I=\int{\cos{bx}.e^{ax}dx}\)
\(\Rightarrow I=\cos{bx}\int{e^{ax}dx}-\int{\left\{\frac{d}{dx}(\cos{bx})\int{e^{ax}dx}\right\}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(\Rightarrow I=\cos{bx}.\frac{e^{ax}}{a}-\int{\left\{(-\sin{bx}).\frac{d}{dx}(bx).\frac{e^{ax}}{a}\right\}dx}\) ➜Note \(\because \int{e^{ax}dx}=\frac{e^{ax}}{a}, \frac{d}{dx}(\cos{x})=-\sin{x}\)
\(\Rightarrow I=\frac{1}{a}e^{ax}\cos{bx}+\int{\left\{\sin{bx}.b.\frac{e^{ax}}{a}\right\}dx}\)
\(\Rightarrow I=\frac{1}{a}e^{ax}\cos{bx}+\frac{b}{a}\int{\sin{bx}.e^{ax}dx}\)
\(\Rightarrow I=\frac{1}{a}e^{ax}\cos{bx}+\frac{b}{a}\left[\sin{bx}\int{e^{ax}dx}-\int{\left\{\frac{d}{dx}(\sin{bx})\int{e^{ax}dx}\right\}dx}\right]\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(\Rightarrow I=\frac{1}{a}e^{ax}\cos{bx}+\frac{b}{a}\left[\sin{bx}.\frac{e^{ax}}{a}-\int{\cos{bx}.\frac{d}{dx}(bx).\frac{e^{ax}}{a}dx}\right]\) ➜Note \(\because \int{e^{ax}dx}=\frac{e^{ax}}{a}, \frac{d}{dx}(\sin{x})=\cos{x}\)
\(\Rightarrow I=\frac{1}{a}e^{ax}\cos{bx}+\frac{b}{a}\left[\frac{1}{a}e^{ax}\sin{bx}-\int{\cos{bx}.b.\frac{e^{ax}}{a}dx}\right]\)
\(\Rightarrow I=\frac{1}{a}e^{ax}\cos{bx}+\frac{b}{a}\left[\frac{1}{a}e^{ax}\sin{bx}-\frac{b}{a}\int{\cos{bx}.e^{ax}dx}\right]\)
\(\Rightarrow I=\frac{1}{a}e^{ax}\cos{bx}+\frac{b}{a^2}e^{ax}\sin{bx}-\frac{b^2}{a^2}\int{\cos{bx}.e^{ax}dx}\)
\(\Rightarrow I=\frac{1}{a}e^{ax}\cos{bx}+\frac{b}{a^2}e^{ax}\sin{bx}-\frac{b^2}{a^2}I\) ➜Note \(\because \int{\cos{bx}.e^{ax}dx}=I\)
\(\Rightarrow I+\frac{b^2}{a^2}I=\frac{1}{a}e^{ax}\cos{bx}+\frac{b}{a^2}e^{ax}\sin{bx}\)
\(\Rightarrow I\left(1+\frac{b^2}{a^2}\right)=\frac{1}{a^2}e^{ax}(a\cos{bx}+b\sin{bx})\)
\(\Rightarrow I\left(\frac{a^2+b^2}{a^2}\right)=\frac{1}{a^2}e^{ax}(a\cos{bx}+b\sin{bx})\)
\(\Rightarrow I=\frac{a^2}{a^2+b^2}\times{\frac{1}{a^2}}e^{ax}(a\cos{bx}+b\sin{bx})+c\) ➜Note\(c\) যোগজীকরণ ধ্রুবক।
\(\Rightarrow I=\frac{e^{ax}}{a^2+b^2}(a\cos{bx}+b\sin{bx})+c\)
\(\Rightarrow I=R.S\)
\(\therefore L.S=R.S\)
(Proved)

\(L.S=\int{\sqrt{x^2-a^2}dx}\)
\(=\int{\sqrt{a^2\sec^2{\theta}-a^2}.a\sec{\theta}\tan{\theta}d\theta}\)
\(=\int{\sqrt{a^2(\sec^2{\theta}-1)}.a\sec{\theta}\tan{\theta}d\theta}\)
\(=\int{a\sqrt{\tan^2{\theta}}.a\sec{\theta}\tan{\theta}d\theta}\)
\(=a^2\int{\tan{\theta}.\sec{\theta}\tan{\theta}d\theta}\)
\(=a^2\left[\tan{\theta}\int{\sec{\theta}\tan{\theta}d\theta}-\int{\left\{\frac{d}{d\theta}(\tan{\theta})\int{\sec{\theta}\tan{\theta}d\theta}\right\}d\theta}\right]\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=a^2\left[\tan{\theta}\sec{\theta}-\int{\sec^2{\theta}.\sec{\theta}d\theta}\right]\)
\(=a^2\tan{\theta}\sec{\theta}-a^2\int{\sec^3{\theta}d\theta}\)
\(=a^2\tan{\theta}\sec{\theta}-a^2I\)
ধরি,
\(I=\int{\sec^3{\theta}d\theta}\)
\(\Rightarrow I=\int{\sec^2{\theta}.\sec{\theta}d\theta}\)
\(\Rightarrow I=\sec{\theta}\int{\sec^2{\theta}d\theta}-\int{\left\{\frac{d}{d\theta}(\sec{\theta})\int{\sec^2{\theta}d\theta}\right\}d\theta}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(\Rightarrow I=\sec{\theta}\tan{\theta}-\int{\sec{\theta}\tan{\theta}.\tan{\theta}d\theta}\) ➜Note \(\because \int{\sec^2{x}dx}=\tan{x}, \frac{d}{dx}(sec{x})=sec{x}\tan{x}\)
\(\Rightarrow I=\sec{\theta}\tan{\theta}-\int{\sec{\theta}\tan^2{\theta}d\theta}\)
\(\Rightarrow I=\sec{\theta}\tan{\theta}-\int{\sec{\theta}(\sec^2{\theta}-1)d\theta}\) ➜Note \(\because tan^2{x}=\sec^2{x}-1\)
\(\Rightarrow I=\sec{\theta}\tan{\theta}-\int{\sec^3{\theta}d\theta}+\int{\sec{\theta}d\theta}\)
\(\Rightarrow I=\sec{\theta}\tan{\theta}-I+\int{\sec{\theta}d\theta}\) ➜Note \(\because \int{\sec^3{\theta}d\theta}=I\)
\(\Rightarrow I+I=\sec{\theta}\tan{\theta}+\ln{\left|\sec{\theta}+\tan{\theta}\right|}\)
\(\Rightarrow 2I=\sec{\theta}\tan{\theta}+\ln{\left|\sec{\theta}+\tan{\theta}\right|}\)
\(\Rightarrow I=\frac{1}{2}\left(\sec{\theta}\tan{\theta}+\ln{\left|\sec{\theta}+\tan{\theta}\right|}\right)+c_{1}\) ➜Note \(\because \int{\sec{\theta}d\theta}=\ln{\left|\sec{x}+\tan{x}\right|}\) এবং \(c_{1}\) যোগজীকরণ ধ্রুবক।
এখন,
\(L.S=a^2\tan{\theta}\sec{\theta}-a^2I\)
\(=a^2\tan{\theta}\sec{\theta}-a^2\left\{\frac{1}{2}\left(\sec{\theta}\tan{\theta}+\ln{\left|\sec{\theta}+\tan{\theta}\right|}\right)+c_{1}\right\}\)
\(=a^2\tan{\theta}\sec{\theta}-\frac{a^2}{2}\left(\sec{\theta}\tan{\theta}+\ln{\left|\sec{\theta}+\tan{\theta}\right|}\right)-a^2c_{1}\)
\(=a^2\tan{\theta}\sec{\theta}-\frac{a^2}{2}\sec{\theta}\tan{\theta}-\frac{a^2}{2}\ln{\left|\sec{\theta}+\tan{\theta}\right|}-a^2c_{1}\)
\(=\frac{2a^2-a^2}{2}\sec{\theta}\tan{\theta}-\frac{a^2}{2}\ln{\left|\sec{\theta}+\tan{\theta}\right|}-a^2c_{1}\)
\(=\frac{a^2}{2}\sec{\theta}\sqrt{\sec^2{\theta}-1}-\frac{a^2}{2}\ln{\left|\sec{\theta}+\sqrt{\sec^2{\theta}-1}\right|}-a^2c_{1}\) ➜Note \(\because \tan{x}=\sqrt{\sec^2{x}-1}\)
\(=\frac{a^2}{2}\times{\frac{x}{a}}\sqrt{\left(\frac{x}{a}\right)^2-1}-\frac{a^2}{2}\ln{\left|\frac{x}{a}+\sqrt{\left(\frac{x}{a}\right)^2-1}\right|}-a^2c_{1}\) ➜Note \(\because \sec{\theta}=\frac{x}{a}\)
\(=\frac{ax}{2}.\sqrt{\frac{x^2}{a^2}-1}-\frac{a^2}{2}\ln{\left|\frac{x}{a}+\sqrt{\frac{x^2}{a^2}-1}\right|}-a^2c_{1}\)
\(=\frac{ax}{2}.\sqrt{\frac{x^2-a^2}{a^2}}-\frac{a^2}{2}\ln{\left|\frac{x}{a}+\sqrt{\frac{x^2-a^2}{a^2}}\right|}-a^2c_{1}\)
\(=\frac{ax}{2}.\frac{\sqrt{x^2-a^2}}{a}-\frac{a^2}{2}\ln{\left|\frac{x}{a}+\frac{\sqrt{x^2-a^2}}{a}\right|}-a^2c_{1}\)
\(=\frac{x\sqrt{x^2-a^2}}{2}-\frac{a^2}{2}\ln{\left|\frac{x+\sqrt{x^2-a^2}}{a}\right|}-a^2c_{1}\)
\(=\frac{x\sqrt{x^2-a^2}}{2}-\frac{a^2}{2}\ln{\frac{\left|x+\sqrt{x^2-a^2}\right|}{|a|}}-a^2c_{1}\)
\(=\frac{x\sqrt{x^2-a^2}}{2}-\frac{a^2}{2}\left(\ln{\left|x+\sqrt{x^2-a^2}\right|}-\ln{|a|}\right)-a^2c_{1}\) ➜Note \(\because \ln{\frac{M}{N}}=\ln{M}-\ln{N}\)
\(=\frac{x\sqrt{x^2-a^2}}{2}-\frac{a^2}{2}\ln{\left|x+\sqrt{x^2-a^2}\right|}+\frac{a^2}{2}\ln{|a|}-a^2c_{1}\)
\(=\frac{x\sqrt{x^2-a^2}}{2}-\frac{a^2}{2}\ln{\left|x+\sqrt{x^2-a^2}\right|}+c\) ➜Note \(\because c=\frac{a^2}{2}\ln{|a|}-a^2c_{1}\)
\(=R.S\)
\(\therefore L.S=R.S\)
(Proved)

\(L.S=\int{\sqrt{x^2+a^2}dx}\)
\(=\int{\sqrt{a^2\tan^2{\theta}+a^2}.a\sec^2{\theta}d\theta}\)
\(=\int{\sqrt{a^2(\tan^2{\theta}+1)}.a\sec^2{\theta}d\theta}\)
\(=\int{a\sqrt{\sec^2{\theta}}.a\sec^2{\theta}d\theta}\)
\(=a^2\int{\sec{\theta}.\sec^2{\theta}d\theta}\)
\(=a^2\int{\sec^3{\theta}d\theta}\)
\(=a^2I\)
ধরি,
\(I=\int{\sec^3{\theta}d\theta}\)
\(\Rightarrow I=\int{\sec^2{\theta}.\sec{\theta}d\theta}\)
\(\Rightarrow I=\sec{\theta}\int{\sec^2{\theta}d\theta}-\int{\left\{\frac{d}{d\theta}(\sec{\theta})\int{\sec^2{\theta}d\theta}\right\}d\theta}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(\Rightarrow I=\sec{\theta}\tan{\theta}-\int{\sec{\theta}\tan{\theta}.\tan{\theta}d\theta}\) ➜Note \(\because \int{\sec^2{x}dx}=\tan{x}, \frac{d}{dx}(sec{x})=sec{x}\tan{x}\)
\(\Rightarrow I=\sec{\theta}\tan{\theta}-\int{\sec{\theta}\tan^2{\theta}d\theta}\)
\(\Rightarrow I=\sec{\theta}\tan{\theta}-\int{\sec{\theta}(\sec^2{\theta}-1)d\theta}\) ➜Note \(\because tan^2{x}=\sec^2{x}-1\)
\(\Rightarrow I=\sec{\theta}\tan{\theta}-\int{\sec^3{\theta}d\theta}+\int{\sec{\theta}d\theta}\)
\(\Rightarrow I=\sec{\theta}\tan{\theta}-I+\int{\sec{\theta}d\theta}\) ➜Note \(\because \int{\sec^3{\theta}d\theta}=I\)
\(\Rightarrow I+I=\sec{\theta}\tan{\theta}+\ln{\left|\sec{\theta}+\tan{\theta}\right|}\)
\(\Rightarrow 2I=\sec{\theta}\tan{\theta}+\ln{\left|\sec{\theta}+\tan{\theta}\right|}\)
\(\Rightarrow I=\frac{1}{2}\left(\sec{\theta}\tan{\theta}+\ln{\left|\sec{\theta}+\tan{\theta}\right|}\right)+c_{1}\) ➜Note \(\because \int{\sec{\theta}d\theta}=\ln{\left|\sec{x}+\tan{x}\right|}\) এবং \(c_{1}\) যোগজীকরণ ধ্রুবক।
এখন,
\(L.S=a^2I\)
\(=a^2\left\{\frac{1}{2}\left(\sec{\theta}\tan{\theta}+\ln{\left|\sec{\theta}+\tan{\theta}\right|}\right)+c_{1}\right\}\)
\(=\frac{a^2}{2}\left(\sec{\theta}\tan{\theta}+\ln{\left|\sec{\theta}+\tan{\theta}\right|}\right)-a^2c_{1}\)
\(=\frac{a^2}{2}\sec{\theta}\tan{\theta}+\frac{a^2}{2}\ln{\left|\tan{\theta}+\sec{\theta}\right|}-a^2c_{1}\)
\(=\frac{a^2}{2}\tan{\theta}\sqrt{\tan^2{\theta}+1}+\frac{a^2}{2}\ln{\left|\tan{\theta}+\sqrt{\tan^2{\theta}+1}\right|}-a^2c_{1}\) ➜Note \(\because \sec{x}=\sqrt{\tan^2{x}+1}\)
\(=\frac{a^2}{2}\times{\frac{x}{a}}\sqrt{\left(\frac{x}{a}\right)^2+1}+\frac{a^2}{2}\ln{\left|\frac{x}{a}+\sqrt{\left(\frac{x}{a}\right)^2+1}\right|}-a^2c_{1}\) ➜Note \(\because \tan{\theta}=\frac{x}{a}\)
\(=\frac{ax}{2}.\sqrt{\frac{x^2}{a^2}+1}+\frac{a^2}{2}\ln{\left|\frac{x}{a}+\sqrt{\frac{x^2}{a^2}+1}\right|}-a^2c_{1}\)
\(=\frac{ax}{2}.\sqrt{\frac{x^2+a^2}{a^2}}+\frac{a^2}{2}\ln{\left|\frac{x}{a}+\sqrt{\frac{x^2+a^2}{a^2}}\right|}-a^2c_{1}\)
\(=\frac{ax}{2}.\frac{\sqrt{x^2+a^2}}{a}+\frac{a^2}{2}\ln{\left|\frac{x}{a}+\frac{\sqrt{x^2+a^2}}{a}\right|}-a^2c_{1}\)
\(=\frac{x\sqrt{x^2+a^2}}{2}+\frac{a^2}{2}\ln{\left|\frac{x+\sqrt{x^2+a^2}}{a}\right|}-a^2c_{1}\)
\(=\frac{x\sqrt{x^2+a^2}}{2}+\frac{a^2}{2}\ln{\frac{\left|x+\sqrt{x^2+a^2}\right|}{|a|}}-a^2c_{1}\)
\(=\frac{x\sqrt{x^2+a^2}}{2}+\frac{a^2}{2}\left(\ln{\left|x+\sqrt{x^2+a^2}\right|}-\ln{|a|}\right)-a^2c_{1}\) ➜Note \(\because \ln{\frac{M}{N}}=\ln{M}-\ln{N}\)
\(=\frac{x\sqrt{x^2+a^2}}{2}+\frac{a^2}{2}\ln{\left|x+\sqrt{x^2+a^2}\right|}-\frac{a^2}{2}\ln{|a|}-a^2c_{1}\)
\(=\frac{x\sqrt{x^2+a^2}}{2}+\frac{a^2}{2}\ln{\left|x+\sqrt{x^2+a^2}\right|}+c\) ➜Note \(\because c=-\frac{a^2}{2}\ln{|a|}-a^2c_{1}\)
\(=R.S\)
\(\therefore L.S=R.S\)
(Proved)

\(L.S=\int{e^x\{f(x)+f^{\prime}(x)\}dx}\)
\(=\int{\{e^xf(x)+e^xf^{\prime}(x)\}dx}\)
\(=\int{e^xf(x)dx}+\int{e^xf^{\prime}(x)dx}\)
\(=\int{e^xf(x)dx}+e^x\int{f^{\prime}(x)dx}-\int{\left\{\frac{d}{dx}(e^x)\int{f^{\prime}(x)dx}\right\}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\int{e^xf(x)dx}+e^xf(x)-\int{e^xf(x)dx}\) ➜Note \(\because \int{f^{\prime}(x)dx}=f(x), \frac{d}{dx}(e^x)=e^x\)
\(=e^xf(x)+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।
\(=R.S\)
\(\therefore L.S=R.S\)
(Proved)

\(L.S=\int{e^{ax}\{af(x)+f^{\prime}(x)\}dx}\)
\(=\int{\{ae^{ax}f(x)+e^{ax}f^{\prime}(x)\}dx}\)
\(=\int{ae^{ax}f(x)dx}+\int{e^{ax}f^{\prime}(x)dx}\)
\(=\int{ae^{ax}f(x)dx}+e^{ax}\int{f^{\prime}(x)dx}-\int{\left\{\frac{d}{dx}(e^{ax})\int{f^{\prime}(x)dx}\right\}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\int{ae^{ax}f(x)dx}+e^{ax}f(x)-\int{e^{ax}.\frac{d}{dx}(ax).f(x)dx}\) ➜Note \(\because \int{f^{\prime}(x)dx}=f(x), \frac{d}{dx}(e^x)=e^x\)
\(=\int{ae^{ax}f(x)dx}+e^{ax}f(x)-\int{e^{ax}.a.f(x)dx}\)
\(=\int{ae^{ax}f(x)dx}+e^{ax}f(x)-\int{ae^{ax}f(x)dx}\)
\(=e^{ax}f(x)+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।
\(=R.S\)
\(\therefore L.S=R.S\)
(Proved)

\(\int{x^2\ln{|x|}dx}\)
\(=\int{\ln{|x|}.x^2dx}\)
\(=\ln{|x|}\int{x^2dx}-\int{\left\{\frac{d}{dx}(\ln{|x|})\int{x^2dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\ln{|x|}\frac{x^3}{3}-\int{\left\{\frac{1}{x}.\frac{x^3}{3}\right\}dx}\) ➜Note \(\because \frac{d}{dx}(\ln{|x|})=\frac{1}{x}, \int{x^2dx}=\frac{x^3}{3}\)
\(=\frac{1}{3}x^3\ln{|x|}-\int{\frac{x^2}{3}dx}\)
\(=\frac{1}{3}x^3\ln{|x|}-\frac{1}{3}\int{x^2dx}\)
\(=\frac{1}{3}x^3\ln{|x|}-\frac{1}{3}.\frac{x^3}{3}+c\) ➜Note \(\because \int{x^2dx}=\frac{x^3}{3}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{3}x^3\ln{|x|}-\frac{1}{9}x^3+c\)
\(=\frac{1}{9}x^3\left(3\ln{|x|}-1\right)+c\)

\(\int{e^x\sin{x}dx}\)
\(=\int{\sin{x}.e^xdx}\)
ধরি,
\(I=\int{\sin{x}.e^xdx}\)
\(=\sin{x}\int{e^xdx}-\int{\left\{\frac{d}{dx}(\sin{x})\int{e^xdx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\sin{x}.e^x-\int{\cos{x}.e^xdx}\) ➜Note \(\because \int{e^xdx}=e^x, \frac{d}{dx}(\sin{x})=\cos{x}\)
\(=e^x\sin{x}-\left[\cos{x}\int{e^xdx}-\int{\left\{\frac{d}{dx}(\cos{x})\int{e^xdx}\right\}dx}\right]\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=e^x\sin{x}-\left[\cos{x}.e^x-\int{(-\sin{x})e^xdx}\right]\) ➜Note \(\because \frac{d}{dx}(\cos{x})=-\sin{x}, \int{e^xdx}=e^x\)
\(=e^x\sin{x}-\left[\cos{x}.e^x+\int{\sin{x}e^xdx}\right]\)
\(\Rightarrow I=e^x\sin{x}-e^x\cos{x}-I\) ➜Note \(\because \int{\sin{x}e^xdx}=I\)
\(\Rightarrow I+I=e^x\sin{x}-e^x\cos{x}\)
\(\Rightarrow 2I=e^x\sin{x}-e^x\cos{x}\)
\(\Rightarrow I=\frac{1}{2}e^x\left(\sin{x}-\cos{x}\right)+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।
\(\therefore \int{\sin{x}e^xdx}=\frac{1}{2}e^x\left(\sin{x}-\cos{x}\right)+c\)

\(\int{\sec^3{x}dx}\)
ধরি,
\(I=\int{\sec^3{x}dx}\)
\(=\int{\sec{x}.\sec^2{x}dx}\)
\(=\sec{x}\int{\sec^2{x}dx}-\int{\left\{\frac{d}{dx}(\sec{x})\int{\sec^2{x}dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\sec{x}\tan{x}-\int{\sec{x}\tan{x}.\tan{x}dx}\) ➜Note \(\because \int{\sec^2dx}=\tan{x}, \frac{d}{dx}(\sec{x})=\sec{x}\tan{x}\)
\(=\sec{x}\tan{x}-\int{\sec{x}\tan^2{x}dx}\)
\(=\sec{x}\tan{x}-\int{\sec{x}(\sec^2{x}-1)dx}\) ➜Note \(\because \tan^2{x}=\sec^2{x}-1\)
\(=\sec{x}\tan{x}-\int{(\sec^3{x}-\sec{x})dx}\)
\(=\sec{x}\tan{x}-\int{\sec^3{x}dx}+\int{\sec{x}dx}\)
\(=\sec{x}\tan{x}-\int{\sec^3{x}dx}+\ln{|\sec{x}+\tan{x}|}+c_{1}\) ➜Note \(\because \int{\sec{x}dx}=\ln{|\sec{x}+\tan{x}|}\), \(c_{1}\) যোগজীকরণ ধ্রুবক।
\(\Rightarrow I=\sec{x}\tan{x}-I+\ln{|\sec{x}+\tan{x}|}+c_{1}\) ➜Note \(\because \int{\sec^3{x}dx}=I\)
\(\Rightarrow I+I=\sec{x}\tan{x}+\ln{|\sec{x}+\tan{x}|}+c_{1}\)
\(\Rightarrow 2I=\sec{x}\tan{x}+\ln{|\sec{x}+\tan{x}|}+c_{1}\)
\(\Rightarrow I=\frac{1}{2}\left(\sec{x}\tan{x}+\ln{|\sec{x}+\tan{x}|}\right)+\frac{1}{2}c_{1}\)
\(\therefore \int{\sec^3{x}dx}=\frac{1}{2}\left(\sec{x}\tan{x}+\ln{|\sec{x}+\tan{x}|}\right)+c\) ➜Note \(c=\frac{1}{2}c_{1}\) যোগজীকরণ ধ্রুবক।

\(\int{e^x\left(\ln{|x|}+\frac{1}{x}\right)dx}\)
\(=\int{\left(e^x\ln{|x|}+e^x\frac{1}{x}\right)dx}\)
\(=\int{e^x\ln{|x|}dx}+\int{e^x\frac{1}{x}dx}\)
\(=\ln{|x|}\int{e^xdx}-\int{\left\{\frac{d}{dx}(\ln{|x|})\int{e^xdx}\right\}dx}+\int{e^x\frac{1}{x}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\ln{|x|}e^x-\int{\frac{1}{x}.e^xdx}+\int{e^x\frac{1}{x}dx}\) ➜Note \(\because \int{e^xdx}=e^x, \frac{d}{dx}(\ln{|x|})=\frac{1}{x}\)
\(=\ln{|x|}e^x-\int{e^x\frac{1}{x}dx}+\int{e^x\frac{1}{x}dx}\)
\(=e^x\ln{|x|}+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।

\(\int{xe^xdx}\)
\(=x\int{e^xdx}-\int{\left\{\frac{d}{dx}(x)\int{e^xdx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=xe^x-\int{1.e^xdx}\) ➜Note \(\because \int{e^xdx}=e^x, \frac{d}{dx}(x)=1\)
\(=xe^x-\int{e^xdx}\)
\(=xe^x-e^x+c\) ➜Note \(\because \int{e^xdx}=e^x\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=e^x(x-1)+c\)

\(\int{x\cos{x}dx}\)
\(=x\int{\cos{x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\cos{x}dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=x\sin{x}-\int{1.\sin{x}dx}\) ➜Note \(\because \int{\cos{x}dx}=\sin{x}, \frac{d}{dx}(x)=1\)
\(=x\sin{x}-\int{\sin{x}dx}\)
\(=x\sin{x}+\cos{x}+c\) ➜Note \(\because \int{\sin{x}dx}=-\cos{x}\) এবং \(c\) যোগজীকরণ ধ্রুবক।

\(\int{x\ln{|x|}dx}\)
\(=\int{\ln{|x|}.xdx}\)
\(=\ln{|x|}\int{xdx}-\int{\left\{\frac{d}{dx}(\ln{|x|})\int{xdx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\ln{|x|}.\frac{x^2}{2}-\int{\frac{1}{x}.\frac{x^2}{2}dx}\) ➜Note \(\because \int{xdx}=\frac{x^2}{2}, \frac{d}{dx}(\ln{|x|})=\frac{1}{x}\)
\(=\frac{1}{2}x^2\ln{|x|}-\frac{1}{2}\int{xdx}\)
\(=\frac{1}{2}x^2\ln{|x|}-\frac{1}{2}.\frac{x^2}{2}+c\) ➜Note \(\because \int{xdx}=\frac{x^2}{2}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{2}x^2\ln{|x|}-\frac{1}{4}x^2+c\)
\(=\frac{1}{4}x^2(2\ln{|x|}-1)+c\)

\(\int{x\tan^{-1}{x}dx}\)
\(=\tan^{-1}{x}\int{xdx}-\int{\left\{\frac{d}{dx}(\tan^{-1}{x})\int{xdx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\tan^{-1}{x}.\frac{x^2}{2}-\int{\frac{1}{1+x^2}.\frac{x^2}{2}dx}\) ➜Note \(\because \int{xdx}=\frac{x^2}{2}, \frac{d}{dx}(\tan^{-1}{x})=\frac{1}{1+x^2}\)
\(=\frac{1}{2}x^2\tan^{-1}{x}-\frac{1}{2}\int{\frac{x^2}{1+x^2}dx}\)
\(=\frac{1}{2}x^2\tan^{-1}{x}-\frac{1}{2}\int{\frac{1+x^2-1}{1+x^2}dx}\)
\(=\frac{1}{2}x^2\tan^{-1}{x}-\frac{1}{2}\int{\left(\frac{1+x^2}{1+x^2}-\frac{1}{1+x^2}\right)dx}\)
\(=\frac{1}{2}x^2\tan^{-1}{x}-\frac{1}{2}\int{\left(1-\frac{1}{1+x^2}\right)dx}\)
\(=\frac{1}{2}x^2\tan^{-1}{x}-\frac{1}{2}\int{1dx}+\frac{1}{2}\int{\frac{1}{1+x^2}dx}\)
\(=\frac{1}{2}x^2\tan^{-1}{x}-\frac{1}{2}x+\frac{1}{2}\tan^{-1}{x}+c\) ➜Note \(\because \int{1dx}=x, \int{\frac{1}{1+x^2}dx}=\tan^{-1}{x}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{2}(x^2\tan^{-1}{x}+\tan^{-1}{x}-x)+c\)

\(\int{e^x(\cos{x}+\sin{x})dx}\)
\(=\int{e^{x}(\sin{x}+\cos{x})dx}\)
\(=\int{e^{x}\sin{x}dx}+\int{e^{x}\cos{x})dx}\)
\(=\sin{x}\int{e^{x}dx}-\int{\left\{\frac{d}{dx}(\sin{x})\int{e^xdx}\right\}dx}+\int{e^{x}\cos{x})dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\sin{x}.e^{x}-\int{\cos{x}.e^xdx}+\int{e^{x}\cos{x})dx}\) ➜Note \(\because \int{e^xdx}=e^x, \frac{d}{dx}(\sin{x})=\cos{x}\)
\(=e^{x}\sin{x}-\int{e^{x}\cos{x}dx}+\int{e^{x}\cos{x})dx}\)
\(=e^{x}\sin{x}+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।

\(\int{x^3\sin{2x}dx}\)
\(=x^3\int{\sin{2x}dx}-\int{\left\{\frac{d}{dx}(x^3)\int{\sin{2x}dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=x^3\left(-\frac{\cos{2x}}{2}\right)-\int{3x^2.\left(-\frac{\cos{2x}}{2}\right)dx}\) ➜Note \(\because \int{\sin{ax}dx}=-\frac{\cos{ax}}{a}, \frac{d}{dx}(x^3)=3x^2\)
\(=-\frac{1}{2}x^3\cos{2x}+\frac{3}{2}\int{x^2\cos{2x}dx}\)
\(=-\frac{1}{2}x^3\cos{2x}+\frac{3}{2}\left[x^2\int{\cos{2x}dx}-\int{\left\{\frac{d}{dx}(x^2)\int{\cos{2x}dx}\right\}dx}\right]\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=-\frac{1}{2}x^3\cos{2x}+\frac{3}{2}\left[x^2.\frac{\sin{2x}}{2}-\int{2x.\frac{\sin{2x}}{2}dx}\right]\) ➜Note \(\because \int{\cos{ax}dx}=.\frac{\sin{ax}}{a}, \frac{d}{dx}(x^2)=2x\)
\(=-\frac{1}{2}x^3\cos{2x}+\frac{3}{4}x^2\sin{2x}-\frac{3}{4}\int{x\sin{2x}dx}\)
\(=-\frac{1}{2}x^3\cos{2x}+\frac{3}{4}x^2\sin{2x}-\frac{3}{4}\left[x\int{\sin{2x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\sin{2x}dx}\right\}dx}\right]\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=-\frac{1}{2}x^3\cos{2x}+\frac{3}{4}x^2\sin{2x}-\frac{3}{4}\left[x\left(-\frac{\cos{2x}}{2}\right)-\int{1.\left(-\frac{\cos{2x}}{2}\right)dx}\right]\) ➜Note \(\because \int{\sin{ax}dx}=-\frac{\cos{ax}}{a}, \frac{d}{dx}(x)=1\)
\(=-\frac{1}{2}x^3\cos{2x}+\frac{3}{4}x^2\sin{2x}-\frac{3}{4}\left[-\frac{1}{2}x\cos{2x}+\frac{1}{2}\int{\cos{2x}dx}\right]\)
\(=-\frac{1}{2}x^3\cos{2x}+\frac{3}{4}x^2\sin{2x}+\frac{3}{8}x\cos{x}-\frac{3}{8}\int{\cos{2x}dx}\)
\(=-\frac{1}{2}x^3\cos{2x}+\frac{3}{4}x^2\sin{2x}+\frac{3}{8}x\cos{x}-\frac{3}{8}\frac{\sin{2x}}{2}+c\) ➜Note \(\because \int{\cos{ax}dx}=\frac{\sin{ax}}{a}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{3}{4}x^2\sin{2x}+\frac{3}{8}x\cos{x}-\frac{1}{2}x^3\cos{2x}-\frac{3}{16}\sin{2x}+c\)
\(=\frac{1}{16}(12x^2\sin{2x}+6x\cos{x}-8x^3\cos{2x}-3\sin{2x})+c\)

\(\int{x\sin{x}dx}\)
\(=x\int{\sin{x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\sin{x}dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=x(-\cos{x})-\int{1.(-\cos{x})dx}\) ➜Note \(\because \int{\sin{x}dx}=-\cos{x}, \frac{d}{dx}(x)=1\)
\(=-x\cos{x}+\int{\cos{x}dx}\)
\(=-x\cos{x}+\sin{x}+c\) ➜Note \(\because \int{\cos{x}dx}=\sin{x}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\sin{x}-x\cos{x}+c\)

\(\int{x\cos{x}dx}\)
\(=x\int{\cos{x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\cos{x}dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=x\sin{x}-\int{1.\sin{x}dx}\) ➜Note \(\because \int{\cos{x}dx}=\sin{x}, \frac{d}{dx}(x)=1\)
\(=x\sin{x}-\int{\sin{x}dx}\)
\(=x\sin{x}+\cos{x}+c\) ➜Note \(\because \int{\sin{x}dx}=-\cos{x}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=x\sin{x}+\cos{x}+c\)

\(\int{x\sin{3x}dx}\)
\(=x\int{\sin{3x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\sin{3x}dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=x\left(-\frac{1}{3}\cos{3x}\right)-\int{1.\left(-\frac{1}{3}\cos{3x}\right)dx}\) ➜Note \(\because \int{\sin{ax}dx}=-\frac{1}{a}\cos{ax}, \frac{d}{dx}(x)=1\)
\(=-\frac{1}{3}x\cos{3x}+\frac{1}{3}\int{\cos{3x}dx}\)
\(=-\frac{1}{3}x\cos{3x}+\frac{1}{3}.\frac{1}{3}\sin{3x}+c\) ➜Note \(\because \int{\cos{ax}dx}=\frac{1}{a}\sin{ax}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=-\frac{1}{3}x\cos{3x}+\frac{1}{9}\sin{3x}+c\)
\(=\frac{1}{9}\sin{3x}-\frac{1}{3}x\cos{3x}+c\)

\(\int{x^2\sin{x}dx}\)
\(=x^2\int{\sin{x}dx}-\int{\left\{\frac{d}{dx}(x^2)\int{\sin{x}dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=x^2(-\cos{x})-\int{2x.(-\cos{x})dx}\) ➜Note \(\because \int{\sin{x}dx}=-\cos{x}, \frac{d}{dx}(x^2)=2x\)
\(=-x^2\cos{x}+2\int{x\cos{x}dx}\)
\(=-x^2\cos{x}+2\left[x\int{\cos{x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\cos{x}dx}\right\}dx}\right]\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=-x^2\cos{x}+2\left[x\sin{x}-\int{1.\sin{x}dx}\right]\) ➜Note \(\because \int{\cos{x}dx}=\sin{x}, \frac{d}{dx}(x)=1\)
\(=-x^2\cos{x}+2x\sin{x}-2\int{\sin{x}dx}\)
\(=-x^2\cos{x}+2x\sin{x}+2\cos{x}+c\) ➜Note \(\because \int{\sin{x}dx}=-\cos{x}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=2x\sin{x}+2\cos{x}-x^2\cos{x}+c\)

\(\int{x^3\sin{x}dx}\)
\(=x^3\int{\sin{x}dx}-\int{\left\{\frac{d}{dx}(x^3)\int{\sin{x}dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=x^3(-\cos{x})-\int{3x^2.(-\cos{x})dx}\) ➜Note \(\because \int{\sin{x}dx}=-\cos{x}, \frac{d}{dx}(x^3)=3x^2\)
\(=-x^3\cos{x}+3\int{x^2\cos{x}dx}\)
\(=-x^3\cos{x}+3\left[x^2\int{\cos{x}dx}-\int{\left\{\frac{d}{dx}(x^2)\int{\cos{x}dx}\right\}dx}\right]\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=-x^3\cos{x}+3\left[x^2\sin{x}-\int{2x.\sin{x}dx}\right]\) ➜Note \(\because \int{\cos{x}dx}=\sin{x}, \frac{d}{dx}(x^2)=2x\)
\(=-x^3\cos{x}+3x^2\sin{x}-6\int{x\sin{x}dx}\)
\(=-x^3\cos{x}+3x^2\sin{x}-6\left[x\int{\sin{x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\sin{x}dx}\right\}dx}\right]\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=-x^3\cos{x}+3x^2\sin{x}-6\left[-x\cos{x}-\int{1.(-\cos{x})dx}\right]\) ➜Note \(\because \int{\sin{x}dx}=-\cos{x}, \frac{d}{dx}(x)=1\)
\(=-x^3\cos{x}+3x^2\sin{x}-6\left[-x\cos{x}+\int{\cos{x}dx}\right]\)
\(=-x^3\cos{x}+3x^2\sin{x}+6x\cos{x}-6\int{\cos{x}dx}\)
\(=3x^2\sin{x}+6x\cos{x}-x^3\cos{x}-6\sin{x}+c\) ➜Note \(\because \int{\cos{x}dx}=\sin{x}\) এবং \(c\) যোগজীকরণ ধ্রুবক।

\(\int{xe^{-x}dx}\)
\(=x\int{e^{-x}dx}-\int{\left\{\frac{d}{dx}(x)\int{e^{-x}dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=x\frac{1}{-1}e^{-x}-\int{1.\frac{1}{-1}e^{-x}dx}\) ➜Note \(\because \int{e^{ax}dx}=\frac{1}{a}e^{ax}, \frac{d}{dx}(x)=1\)
\(=-xe^{-x}+\int{e^{-x}dx}\)
\(=-xe^{-x}+\frac{1}{-1}e^{-x}+c\) ➜Note \(\because \int{e^{ax}dx}=\frac{1}{a}e^{ax}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=-xe^{-x}-e^{-x}+c\)
\(=-(x+1)e^{-x}+c\)

\(\int{x^3e^{2x}dx}\)
\(=x^3\int{e^{2x}dx}-\int{\left\{\frac{d}{dx}(x^3)\int{e^{2x}dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=x^3\frac{1}{2}e^{2x}-\int{3x^2.\frac{1}{2}e^{2x}dx}\) ➜Note \(\because \int{e^{ax}dx}=\frac{1}{a}e^{ax}, \frac{d}{dx}(x^3)=3x^2\)
\(=\frac{1}{2}x^3e^{2x}-\frac{3}{2}\int{x^2e^{2x}dx}\)
\(=\frac{1}{2}x^3e^{2x}-\frac{3}{2}\left[x^2\int{e^{2x}dx}-\int{\left\{\frac{d}{dx}(x^2)\int{e^{2x}dx}\right\}dx}\right]\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{1}{2}x^3e^{2x}-\frac{3}{2}\left[x^2\frac{1}{2}e^{2x}-\int{2x.\frac{1}{2}e^{2x}dx}\right]\) ➜Note \(\because \int{e^{ax}dx}=\frac{1}{a}e^{ax}, \frac{d}{dx}(x^2)=2x\)
\(=\frac{1}{2}x^3e^{2x}-\frac{3}{4}x^2e^{2x}+\frac{3}{2}\int{xe^{2x}dx}\)
\(=\frac{1}{2}x^3e^{2x}-\frac{3}{4}x^2e^{2x}+\frac{3}{2}\left[x\int{e^{2x}dx}-\int{\left\{\frac{d}{dx}(x)\int{e^{2x}dx}\right\}dx}\right]\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{1}{2}x^3e^{2x}-\frac{3}{4}x^2e^{2x}+\frac{3}{2}\left[x\frac{1}{2}e^{2x}-\int{1.\frac{1}{2}e^{2x}dx}\right]\) ➜Note \(\because \int{e^{ax}dx}=\frac{1}{a}e^{ax}, \frac{d}{dx}(x)=1\)
\(=\frac{1}{2}x^3e^{2x}-\frac{3}{4}x^2e^{2x}+\frac{3}{4}xe^{2x}-\frac{3}{4}\int{e^{2x}dx}\)
\(=\frac{1}{2}x^3e^{2x}-\frac{3}{4}x^2e^{2x}+\frac{3}{4}xe^{2x}-\frac{3}{4}.\frac{1}{2}e^{2x}+c\) ➜Note \(\because \int{e^{ax}dx}=\frac{1}{a}e^{ax}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{2}x^3e^{2x}-\frac{3}{4}x^2e^{2x}+\frac{3}{4}xe^{2x}-\frac{3}{8}e^{2x}+c\)
\(=\frac{1}{8}e^{2x}(4x^3-6x^2+6x-3)+c\)

\(\int{x^2e^{x}dx}\)
\(=x^2\int{e^{x}dx}-\int{\left\{\frac{d}{dx}(x^2)\int{e^{x}dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=x^2e^{x}-\int{2x.e^{x}dx}\) ➜Note \(\because \int{e^{x}dx}=e^{x}, \frac{d}{dx}(x^2)=2x\)
\(=x^2e^{x}-2\int{xe^{x}dx}\)
\(=x^2e^{x}-2\left[x\int{e^{x}dx}-\int{\left\{\frac{d}{dx}(x)\int{e^{x}dx}\right\}dx}\right]\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=x^2e^{x}-2\left[xe^x-\int{1.e^xdx}\right]\) ➜Note \(\because \int{e^{x}dx}=e^{x}, \frac{d}{dx}(x)=1\)
\(=x^2e^{x}-2xe^x+2\int{e^xdx}\)
\(=x^2e^{x}-2xe^x+2e^x+c\) ➜Note \(\because \int{e^{x}dx}=e^{x}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=e^{x}(x^2-2x+2)+c\)

\(\int{x^2e^{-3x}dx}\)
\(=x^2\int{e^{-3x}dx}-\int{\left\{\frac{d}{dx}(x^2)\int{e^{-3x}dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=x^2\frac{1}{-3}e^{-3x}-\int{2x.\frac{1}{-3}e^{-3x}dx}\) ➜Note \(\because \int{e^{ax}dx}=\frac{1}{a}e^{ax}, \frac{d}{dx}(x^2)=2x\)
\(=-\frac{1}{3}x^2e^{-3x}+\frac{2}{3}\int{xe^{-3x}dx}\)
\(=-\frac{1}{3}x^2e^{-3x}+\frac{2}{3}\left[x\int{e^{-3x}dx}-\int{\left\{\frac{d}{dx}(x)\int{e^{-3x}dx}\right\}dx}\right]\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=-\frac{1}{3}x^2e^{-3x}+\frac{2}{3}\left[x.\frac{1}{-3}e^{-3x}-\int{1.\frac{1}{-3}e^{-3x}dx}\right]\) ➜Note \(\because \int{e^{ax}dx}=\frac{1}{a}e^{ax}, \frac{d}{dx}(x)=1\)
\(=-\frac{1}{3}x^2e^{-3x}+\frac{2}{3}\left[-\frac{1}{3}xe^{-3x}+\frac{1}{3}\int{e^{-3x}dx}\right]\)
\(=-\frac{1}{3}x^2e^{-3x}-\frac{2}{9}xe^{-3x}+\frac{2}{9}\int{e^{-3x}dx}\)
\(=-\frac{1}{3}x^2e^{-3x}-\frac{2}{9}xe^{-3x}+\frac{2}{9}.\frac{1}{-3}e^{-3x}+c\) ➜Note \(\because \int{e^{ax}dx}=\frac{1}{a}e^{ax}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=-\frac{1}{3}x^2e^{-3x}-\frac{2}{9}xe^{-3x}-\frac{2}{27}e^{-3x}+c\)
\(=-\frac{1}{27}e^{-3x}(9x^2+6x+2)+c\)

\(\int{xe^{ax}dx}\)
\(=x\int{e^{ax}dx}-\int{\left\{\frac{d}{dx}(x)\int{e^{ax}dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=x\frac{1}{a}e^{ax}-\int{1.\frac{1}{a}e^{ax}dx}\) ➜Note \(\because \int{e^{ax}dx}=\frac{1}{a}e^{ax}, \frac{d}{dx}(x)=1\)
\(=\frac{1}{a}xe^{ax}-\frac{1}{a}\int{e^{ax}dx}\)
\(=\frac{1}{a}xe^{ax}-\frac{1}{a}.\frac{1}{a}e^{ax}+c\) ➜Note \(\because \int{e^{ax}dx}=\frac{1}{a}e^{ax}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{a}xe^{ax}-\frac{1}{a^2}e^{ax}+c\)
\(=\frac{1}{a^2}e^{ax}(ax-1)+c\)

\(\int{xe^{x}dx}\)
\(=x\int{e^{x}dx}-\int{\left\{\frac{d}{dx}(x)\int{e^{x}dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=xe^{x}-\int{1.e^{x}dx}\) ➜Note \(\because \int{e^{x}dx}=e^{x}, \frac{d}{dx}(x)=1\)
\(=xe^{x}-\int{e^{x}dx}\)
\(=xe^{ax}-e^{x}+c\) ➜Note \(\because \int{e^{x}dx}=e^{x}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=xe^{x}-e^{x}+c\)
\(=e^{x}(x-1)+c\)

\(\int{x^3\ln{|x|}dx}\)
\(=\int{\ln{|x|}(x^3)dx}\)
\(=\ln{|x|}\int{x^3dx}-\int{\left\{\frac{d}{dx}(\ln{|x|})\int{x^3dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\ln{|x|}.\frac{x^4}{4}-\int{\frac{1}{x}.\frac{x^4}{4}dx}\) ➜Note \(\because \int{x^3dx}=\frac{x^4}{4}, \frac{d}{dx}(\ln{|x|})=\frac{1}{x}\)
\(=\frac{1}{4}x^4\ln{|x|}-\frac{1}{4}\int{x^3dx}\)
\(=\frac{1}{4}x^4\ln{|x|}-\frac{1}{4}.\frac{x^4}{4}+c\) ➜Note \(\because \int{x^3dx}=\frac{x^4}{4}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{4}x^4\ln{|x|}-\frac{1}{16}x^4+c\)
\(=\frac{1}{16}x^4(4\ln{|x|}-1)+c\)

\(\int{(\ln{|x|})^2dx}\)
\(=\int{(\ln{|x|})^2.1dx}\)
\(=(\ln{|x|})^2\int{1dx}-\int{\left\{\frac{d}{dx}(\ln{|x|})^2\int{1dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=(\ln{|x|})^2.x-\int{2(\ln{|x|}).\frac{1}{x}.xdx}\) ➜Note \(\because \int{1dx}=x, \frac{d}{dx}(x^2)=2x, \frac{d}{dx}(\ln{|x|})=\frac{1}{x}\)
\(=x(\ln{|x|})^2-2\int{\ln{|x|}.1dx}\) \(=x(\ln{|x|})^2-2\left[\ln{|x|}\int{1dx}-\int{\left\{\frac{d}{dx}(\ln{|x|})\int{1dx}\right\}dx}\right]\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=x(\ln{|x|})^2-2\left[\ln{|x|}x-\int{\frac{1}{x}.xdx}\right]\)
\(=x(\ln{|x|})^2-2\left[\ln{|x|}x-\int{1dx}\right]\)
\(=x(\ln{|x|})^2-2x\ln{|x|}+2\int{1dx}\)
\(=x(\ln{|x|})^2-2x\ln{|x|}+2x+c\) ➜Note \(\because \int{1dx}=x\) এবং \(c\) যোগজীকরণ ধ্রুবক।

\(\int{\frac{\ln{|x|}}{x^2}dx}\)
\(=\int{\ln{|x|}\frac{1}{x^2}dx}\)
\(=\ln{|x|}\int{\frac{1}{x^2}dx}-\int{\left\{\frac{d}{dx}(\ln{|x|})\int{\frac{1}{x^2}dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\ln{|x|}\times{-\frac{1}{x}}-\int{\frac{1}{x}\times{-\frac{1}{x}}dx}\) ➜Note \(\because \int{\frac{1}{x^2}dx}=-\frac{1}{x}, \frac{d}{dx}(\ln{|x|})=\frac{1}{x}\)
\(=-\frac{1}{x}\ln{|x|}+\int{\frac{1}{x^2}dx}\)
\(=-\frac{1}{x}\ln{|x|}-\frac{1}{x}+c\) ➜Note \(\because \int{\frac{1}{x^2}dx}=-\frac{1}{x}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=-\frac{1}{x}(\ln{|x|}+1)+c\)
\(=-\frac{1}{x}(\ln{|x|}+\ln{e})+c\) ➜Note \(\because 1=\ln{e}\)
\(=-\frac{1}{x}\ln{|xe|}+c\) ➜Note\(\because \ln{|M|}+\ln{|N|}=\ln{|MN|}\)

\(\int{\ln{|x|}dx}\)
\(=\int{\ln{|x|}.1dx}\)
\(=\ln{|x|}\int{1dx}-\int{\left\{\frac{d}{dx}(\ln{|x|})\int{1dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\ln{|x|}.x-\int{\frac{1}{x}.xdx}\) ➜Note \(\because \int{1dx}=x, \frac{d}{dx}(\ln{|x|})=\frac{1}{x}\)
\(=x\ln{|x|}-\int{1dx}\)
\(=x\ln{|x|}-x+c\) ➜Note \(\because \int{1dx}=x\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=x(\ln{|x|}-1)+c\)

\(\int{x\ln{|x|}dx}\)
\(=\int{\ln{|x|}.xdx}\)
\(=\ln{|x|}\int{xdx}-\int{\left\{\frac{d}{dx}(\ln{|x|})\int{xdx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\ln{|x|}.\frac{x^2}{2}-\int{\frac{1}{x}.\frac{x^2}{2}dx}\) ➜Note \(\because \int{xdx}=\frac{x^2}{2}, \frac{d}{dx}(\ln{|x|})=\frac{1}{x}\)
\(=\frac{1}{2}x^2\ln{|x|}-\frac{1}{2}\int{xdx}\)
\(=\frac{1}{2}x^2\ln{|x|}-\frac{1}{2}.\frac{x^2}{2}+c\) ➜Note \(\because \int{xdx}=\frac{x^2}{2}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{2}x^2\ln{|x|}-\frac{1}{4}x^2+c\)
\(=\frac{1}{4}x^2(2\ln{|x|}-1)+c\)

\(\int{x^2(\ln{|x|})^2dx}\)
\(=\int{(\ln{|x|})^2x^2dx}\)
\(=(\ln{|x|})^2\int{x^2dx}-\int{\left\{\frac{d}{dx}(\ln{|x|})^2\int{x^2dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=(\ln{|x|})^2.\frac{x^3}{3}-\int{2(\ln{|x|}).\frac{1}{x}.\frac{x^3}{3}dx}\) ➜Note \(\because \int{x^2dx}=\frac{x^3}{3}, \frac{d}{dx}(x^2)=2x, \frac{d}{dx}(\ln{|x|})=\frac{1}{x}\)
\(=\frac{1}{3}x^3(\ln{|x|})^2-\frac{2}{3}\int{\ln{|x|}.x^2dx}\)
\(=\frac{1}{3}x^3(\ln{|x|})^2-\frac{2}{3}\left[\ln{|x|}\int{x^2dx}-\int{\left\{\frac{d}{dx}(\ln{|x|})\int{x^2dx}\right\}dx}\right]\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{1}{3}x^3(\ln{|x|})^2-\frac{2}{3}\left[\ln{|x|}.\frac{x^3}{3}-\int{\left\{\frac{1}{x}.\frac{x^3}{3}\right\}dx}\right]\) ➜Note \(\because \int{x^2dx}=\frac{x^3}{3}, \frac{d}{dx}(\ln{|x|})=\frac{1}{x}\)
\(=\frac{1}{3}x^3(\ln{|x|})^2-\frac{2}{3}\left[\frac{1}{3}x^3\ln{|x|}-\frac{1}{3}\int{x^2dx}\right]\)
\(=\frac{1}{3}x^3(\ln{|x|})^2-\frac{2}{9}x^3\ln{|x|}+\frac{2}{9}\int{x^2dx}\)
\(=\frac{1}{3}x^3(\ln{|x|})^2-\frac{2}{9}x^3\ln{|x|}+\frac{2}{9}.\frac{x^3}{3}+c\) ➜Note \(\because \int{x^2dx}=\frac{x^3}{3}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{3}x^3(\ln{|x|})^2-\frac{2}{9}x^3\ln{|x|}+\frac{2}{27}x^3+c\)
\(=\frac{1}{27}x^3\{9(\ln{|x|})^2-6\ln{|x|}+2\}+c\)

\(\int{x\sqrt{1+x}dx}\)
\(=\int{x(1+x)^{\frac{1}{2}}dx}\)
\(=x\int{(1+x)^{\frac{1}{2}}dx}-\int{\left\{\frac{d}{dx}(x)\int{(1+x)^{\frac{1}{2}}dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=x\frac{(1+x)^{\frac{1}{2}+1}}{\left(\frac{1}{2}+1\right).1}-\int{1.\frac{(1+x)^{\frac{1}{2}+1}}{\left(\frac{1}{2}+1\right).1}dx}\) ➜Note \(\because \int{(ax+b)^{n}dx}=\frac{(ax+b)^{n+1}}{(n+1).a}, \frac{d}{dx}(x)=1\)
\(=x\frac{(1+x)^{\frac{1+2}{2}}}{\frac{1+2}{2}}-\int{\frac{(1+x)^{\frac{1+2}{2}}}{\frac{1+2}{2}}dx}\)
\(=x\frac{(1+x)^{\frac{3}{2}}}{\frac{3}{2}}-\int{\frac{(1+x)^{\frac{3}{2}}}{\frac{3}{2}}dx}\)
\(=\frac{2}{3}x(1+x)^{\frac{3}{2}}-\frac{2}{3}\int{(1+x)^{\frac{3}{2}}dx}\)
\(=\frac{2}{3}x(1+x)^{\frac{3}{2}}-\frac{2}{3}\frac{(1+x)^{\frac{3}{2}+1}}{\left(\frac{3}{2}+1\right).1}+c\) ➜Note \(\because \int{(ax+b)^{n}dx}=\frac{(ax+b)^{n+1}}{(n+1).a}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{2}{3}x(1+x)^{\frac{3}{2}}-\frac{2}{3}\frac{(1+x)^{\frac{3+2}{2}}}{\frac{3+2}{2}}+c\)
\(=\frac{2}{3}x(1+x)^{\frac{3}{2}}-\frac{2}{3}\frac{(1+x)^{\frac{5}{2}}}{\frac{5}{2}}+c\)
\(=\frac{2}{3}x(1+x)^{\frac{3}{2}}-\frac{4}{15}(1+x)^{\frac{5}{2}}+c\)

\(\int{x\sin{2x}dx}\)
\(=x\int{\sin{2x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\sin{2x}dx}\right\}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=x\left(-\frac{\cos{2x}}{2}\right)-\int{1.\left(-\frac{\cos{2x}}{2}\right)dx}\) ➜Note \(\because \int{\sin{ax}dx}=-\frac{\cos{ax}}{a}\)
\(=-\frac{1}{2}x\cos{2x}+\frac{1}{2}\int{\cos{2x}dx}\)
\(=-\frac{1}{2}x\cos{2x}+\frac{1}{2}\int{\cos{2x}dx}\)
\(=-\frac{1}{2}x\cos{2x}+\frac{1}{2}\times{\frac{\sin{2x}}{2}}+c\) ➜Note \(\because \int{\cos{ax}dx}=\frac{\sin{ax}}{a}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=-\frac{1}{2}x\cos{2x}+\frac{1}{4}\sin{2x}+c\)
\(=\frac{1}{4}\sin{2x}-\frac{1}{2}x\cos{2x}+c\)
\(=\frac{1}{4}(\sin{2x}-2x\cos{2x})+c\)

\(\int{x^2\cos{x}dx}\)
\(=x^2\int{\cos{x}dx}-\int{\left\{\frac{d}{dx}(x^2)\int{\cos{x}dx}\right\}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=x^2\sin{x}-\int{2x.\sin{x}dx}\) ➜Note \(\because \frac{d}{dx}(x^2)=2x, \int{\cos{x}dx}=\sin{x}\)
\(=x^2\sin{x}-2\int{x\sin{x}dx}\)
\(=x^2\sin{x}-2\left[x\int{\sin{x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\sin{x}dx}\right\}dx}\right]\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=x^2\sin{x}-2\left[-x\cos{x}-\int{1.(-\cos{x})dx}\right]\) ➜Note \(\because \frac{d}{dx}(x)=1, \int{\sin{x}dx}=-\cos{x}\)
\(=x^2\sin{x}-2\left[-x\cos{x}+\int{\cos{x})dx}\right]\)
\(=x^2\sin{x}+2x\cos{x}-2\int{\cos{x})dx}\)
\(=x^2\sin{x}+2x\cos{x}-2\sin{x}+c\) ➜Note\(\because \int{\cos{x}dx}=\sin{x}\) এবং \(c\) যোগজীকরণ ধ্রুবক।

\(\int{x(\ln{|x|})^2dx}\)
\(=\int{(\ln{|x|})^2.xdx}\)
\(=(\ln{|x|})^2\int{xdx}-\int{\left\{\frac{d}{dx}(\ln{|x|})^2\int{xdx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=(\ln{|x|})^2.\frac{x^2}{2}-\int{2(\ln{|x|}).\frac{1}{x}.\frac{x^2}{2}dx}\) ➜Note \(\because \int{xdx}=\frac{x^2}{2}, \frac{d}{dx}(x^2)=2x, \frac{d}{dx}(\ln{|x|})=\frac{1}{x}\)
\(=\frac{1}{2}x^2(\ln{|x|})^2-\int{\ln{|x|}.xdx}\)
\(=\frac{1}{2}x^2(\ln{|x|})^2-\left[\ln{|x|}\int{xdx}-\int{\left\{\frac{d}{dx}(\ln{|x|})\int{xdx}\right\}dx}\right]\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{1}{2}x^2(\ln{|x|})^2-\left[\ln{|x|}.\frac{x^2}{2}-\int{\frac{1}{x}.\frac{x^2}{2}dx}\right]\) ➜Note \(\because \int{xdx}=\frac{x^2}{2}, \frac{d}{dx}(\ln{|x|})=\frac{1}{x}\)
\(=\frac{1}{2}x^2(\ln{|x|})^2-\left[\frac{1}{2}x^2\ln{|x|}-\frac{1}{2}\int{xdx}\right]\)
\(=\frac{1}{2}x^2(\ln{|x|})^2-\frac{1}{2}x^2\ln{|x|}+\frac{1}{2}\int{xdx}\)
\(=\frac{1}{2}x^2(\ln{|x|})^2-\frac{1}{2}x^2\ln{|x|}+\frac{1}{2}.\frac{x^2}{2}+c\) ➜Note \(\because \int{xdx}=\frac{x^2}{2}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{4}x^2\{2(\ln{|x|})^2-2\ln{|x|}+1\}+c\)

\(\int{\frac{\ln{(\ln{|x|})}}{x}dx}\)
\(=\int{\ln{(\ln{|x|})}.\frac{1}{x}dx}\)
\(=\int{\ln{t}.dt}\)
\(=\int{\ln{t}.1dt}\)
\(=\ln{t}\int{1dt}-\int{\left\{\frac{d}{dx}(\ln{t})\int{1dt}\right\}dt}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\ln{t}.t-\int{\frac{1}{t}.tdt}\) ➜Note \(\because \int{1dx}=x, \frac{d}{dx}(\ln{|x|})=\frac{1}{x}\)
\(=t\ln{t}-\int{1dt}\)
\(=t\ln{t}-t+c\) ➜Note \(\because \int{1dx}=x\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\ln{|x|}\ln{\ln{|x|}}-\ln{|x|}+c\) ➜Note \(\because t=\ln{|x|}\)
\(=\ln{|x|}\{\ln{(\ln{|x|})}-1\}+c\)

\(\int{\frac{x}{\cos^2{x}}dx}\)
\(=\int{x\frac{1}{\cos^2{x}}dx}\)
\(=\int{x\sec^2{x}dx}\)
\(=x\int{\sec^2{x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\sec^2{x}dx}\right\}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=x\tan{x}-\int{1.\tan{x}dx}\) ➜Note \(\because \int{\sec^2{x}dx}=\tan{x}, \frac{d}{dx}(x)=1\)
\(=x\tan{x}-\int{\tan{x}dx}\)
\(=x\tan{x}+\ln{|\cos{x}|}+c\) ➜Note\(\because \int{\tan{x}dx}=-\ln{|\cos{x}|}\) এবং \(c\) যোগজীকরণ ধ্রুবক।

\(\int{x\sec{x}\tan{x}dx}\)
\(=x\int{\sec{x}\tan{x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\sec{x}\tan{x}dx}\right\}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=x\sec{x}-\int{1.\sec{x}dx}\) ➜Note \(\because \int{\sec{x}\tan{x}dx}=\sec{x}, \frac{d}{dx}(x)=1\)
\(=x\sec{x}-\int{\sec{x}dx}\)
\(=x\sec{x}+\ln{|\sec{x}+\tan{x}|}+c\) ➜Note\(\because \int{\sec{x}dx}=\ln{|\sec{x}+\tan{x}|}\) এবং \(c\) যোগজীকরণ ধ্রুবক।

\(\int{\frac{x}{\sin^2{x}}dx}\)
\(=\int{x\frac{1}{\sin^2{x}}dx}\)
\(=\int{x \ cosec^2{x}dx}\)
\(=x\int{ \ cosec^2{x}dx}-\int{\left\{\frac{d}{dx}(x)\int{ \ cosec^2{x}dx}\right\}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=x(-\cot{x})-\int{1.(-\cot{x})dx}\) ➜Note \(\because \int{ \ cosec^2{x}dx}=-\cot{x}, \frac{d}{dx}(x)=1\)
\(=-x\cot{x}+\int{\cot{x}dx}\)
\(=-x\cot{x}+\ln{|\sin{x}|}+c\) ➜Note\(\because \int{\cot{x}dx}=\ln{|\sin{x}|}\) এবং \(c\) যোগজীকরণ ধ্রুবক।

\(\int{x\cos^3{x}dx}\)
\(=\frac{1}{4}\int{x.4\cos^3{x}dx}\)
\(=\frac{1}{4}\int{x(3\cos{x}+\cos{3x})dx}\) ➜Note \(\because 4\cos^3{A}=3\cos{A}+\cos{3A}\)
\(=\frac{1}{4}\int{(3x\cos{x}+x\cos{3x})dx}\)
\(=\frac{3}{4}\int{x\cos{x}dx}+\frac{1}{4}\int{x\cos{3x}dx}\)
\(=\frac{3}{4}\left[x\int{\cos{x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\cos{x}dx}\right\}dx}\right]+\frac{1}{4}\left[x\int{\cos{3x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\cos{3x}dx}\right\}dx}\right]\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{3}{4}\left[x\sin{x}-\int{1.\sin{x}dx}\right]+\frac{1}{4}\left[x\frac{\sin{3x}}{3}-\int{1.\frac{\sin{3x}}{3}dx}\right]\) ➜Note \(\because \int{\cos{x}dx}=\sin{x}, \frac{d}{dx}(x)=1\), \(\int{\cos{ax}dx}=\frac{\sin{ax}}{a}\)
\(=\frac{3}{4}x\sin{x}-\frac{3}{4}\int{\sin{x}dx}+\frac{1}{4}\left[\frac{1}{3}x\sin{3x}-\frac{1}{3}\int{\sin{3x}dx}\right]\)
\(=\frac{3}{4}x\sin{x}+\frac{3}{4}\cos{x}+\frac{1}{12}x\sin{3x}-\frac{1}{12}\int{\sin{3x}dx}\)
\(=\frac{3}{4}x\sin{x}+\frac{3}{4}\cos{x}+\frac{1}{12}x\sin{3x}-\frac{1}{12}\left(-\frac{\cos{3x}}{3}\right)+c\) ➜Note\(\because \int{\sin{ax}dx}=-\frac{\cos{ax}}{a}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{3}{4}x\sin{x}+\frac{3}{4}\cos{x}+\frac{1}{12}x\sin{3x}+\frac{1}{36}\cos{3x}+c\)
\(=\frac{3}{4}(x\sin{x}+\cos{x})+\frac{1}{36}(3x\sin{3x}+\cos{3x})+c\)

\(\int{xe^{ax}dx}\)
\(=x\int{e^{ax}dx}-\int{\left\{\frac{d}{dx}(x)\int{e^{ax}dx}\right\}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=x\frac{e^{ax}}{a}-\int{1.\frac{e^{ax}}{a}dx}\) ➜Note \(\because \int{e^{ax}dx}=\frac{e^{ax}}{a}, \frac{d}{dx}(x)=1\)
\(=\frac{1}{a}xe^{ax}-\frac{1}{a}\int{e^{ax}dx}\)
\(=\frac{1}{a}xe^{ax}-\frac{1}{a}.\frac{e^{ax}}{a}+c\) ➜Note\(\because \int{e^{ax}dx}=\frac{e^{ax}}{a}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{a}xe^{ax}-\frac{1}{a^2}e^{ax}+c\)
\(=\frac{1}{a^2}e^{ax}(ax-1)+c\)

\(\int{\cos^{-1}{x}dx}\)
\(\int{\cos^{-1}{x}.1dx}\)
\(=\cos^{-1}{x}\int{1dx}-\int{\left\{\frac{d}{dx}(\cos^{-1}{x})\int{1dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\cos^{-1}{x}.x-\int{\left(-\frac{1}{\sqrt{1-x^2}}\right).xdx}\) ➜Note \(\because \int{1dx}=x, \frac{d}{dx}(\cos^{-1}{x})=-\frac{1}{\sqrt{1-x^2}}\)
\(=x\cos^{-1}{x}+\int{\frac{x}{\sqrt{1-x^2}}dx}\)
\(=x\cos^{-1}{x}+\int{\frac{1}{\sqrt{1-x^2}}xdx}\)
\(=x\cos^{-1}{x}+\int{\frac{1}{t}\times{-tdt}}\)
\(=x\cos^{-1}{x}-\int{1dt}\)
\(=x\cos^{-1}{x}-t+c\) ➜Note \(\because \int{1dx}=x\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=x\cos^{-1}{x}-\sqrt{1-x^2}+c\) ➜Note \(\because t=\sqrt{1-x^2}\)

\(\int{\sin^{-1}{x}dx}\)
\(\int{\sin^{-1}{x}.1dx}\)
\(=\sin^{-1}{x}\int{1dx}-\int{\left\{\frac{d}{dx}(\sin^{-1}{x})\int{1dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\sin^{-1}{x}.x-\int{\frac{1}{\sqrt{1-x^2}}.xdx}\) ➜Note \(\because \int{1dx}=x, \frac{d}{dx}(\sin^{-1}{x})=\frac{1}{\sqrt{1-x^2}}\)
\(=x\sin^{-1}{x}-\int{\frac{x}{\sqrt{1-x^2}}dx}\)
\(=x\sin^{-1}{x}-\int{\frac{1}{\sqrt{1-x^2}}xdx}\)
\(=x\sin^{-1}{x}-\int{\frac{1}{t}\times{-tdt}}\)
\(=x\sin^{-1}{x}+\int{1dt}\)
\(=x\sin^{-1}{x}+t+c\) ➜Note \(\because \int{1dx}=x\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=x\sin^{-1}{x}+\sqrt{1-x^2}+c\) ➜Note \(\because t=\sqrt{1-x^2}\)

\(\int{x\tan^{-1}{x}dx}\)
\(=\int{\tan^{-1}{x}.xdx}\)
\(=\tan^{-1}{x}\int{xdx}-\int{\left\{\frac{d}{dx}(\tan^{-1}{x})\int{xdx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\tan^{-1}{x}.\frac{x^2}{2}-\int{\frac{1}{1+x^2}.\frac{x^2}{2}dx}\) ➜Note \(\because \int{xdx}=\frac{x^2}{2}, \frac{d}{dx}(\tan^{-1}{x})=\frac{1}{1+x^2}\)
\(=\frac{1}{2}x^2\tan^{-1}{x}-\frac{1}{2}\int{\frac{x^2}{1+x^2}dx}\)
\(=\frac{1}{2}x^2\tan^{-1}{x}-\frac{1}{2}\int{\frac{1+x^2-1}{1+x^2}dx}\)
\(=\frac{1}{2}x^2\tan^{-1}{x}-\frac{1}{2}\int{\left(\frac{1+x^2}{1+x^2}-\frac{1}{1+x^2}\right)dx}\)
\(=\frac{1}{2}x^2\tan^{-1}{x}-\frac{1}{2}\int{\left(1-\frac{1}{1+x^2}\right)dx}\)
\(=\frac{1}{2}x^2\tan^{-1}{x}-\frac{1}{2}\int{1dx}+\frac{1}{2}\int{\frac{1}{1+x^2}dx}\)
\(=\frac{1}{2}x^2\tan^{-1}{x}-\frac{1}{2}x+\frac{1}{2}\tan^{-1}{x}+c\) ➜Note \(\because \int{1dx}=x, \int{\frac{1}{1+x^2}dx}=\tan^{-1}{x}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{2}(x^2+1)\tan^{-1}{x}-\frac{1}{2}x+c\)

\(\int{x\sin^{-1}{x}dx}\)
\(=\int{\sin^{-1}{x}.xdx}\)
\(=\sin^{-1}{x}\int{xdx}-\int{\left\{\frac{d}{dx}(\sin^{-1}{x})\int{xdx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\sin^{-1}{x}.\frac{x^2}{2}-\int{\frac{1}{\sqrt{1-x^2}}.\frac{x^2}{2}dx}\) ➜Note \(\because \int{xdx}=\frac{x^2}{2}, \frac{d}{dx}(\sin^{-1}{x})=\frac{1}{\sqrt{1-x^2}}\)
\(=\frac{1}{2}x^2\sin^{-1}{x}-\frac{1}{2}\int{\frac{x^2}{\sqrt{1-x^2}}dx}\)
\(=\frac{1}{2}x^2\sin^{-1}{x}-\frac{1}{2}\int{\frac{1-(1-x^2)}{\sqrt{1-x^2}}dx}\)
\(=\frac{1}{2}x^2\sin^{-1}{x}-\frac{1}{2}\int{\left(\frac{1}{\sqrt{1-x^2}}-\frac{1-x^2}{\sqrt{1-x^2}}\right)dx}\)
\(=\frac{1}{2}x^2\sin^{-1}{x}-\frac{1}{2}\int{\left(\frac{1}{\sqrt{1-x^2}}-\frac{\sqrt{1-x^2}.\sqrt{1-x^2}}{\sqrt{1-x^2}}\right)dx}\)
\(=\frac{1}{2}x^2\sin^{-1}{x}-\frac{1}{2}\int{\left(\frac{1}{\sqrt{1-x^2}}-\sqrt{1-x^2}\right)dx}\)
\(=\frac{1}{2}x^2\sin^{-1}{x}-\frac{1}{2}\int{\frac{1}{\sqrt{1-x^2}}dx}+\frac{1}{2}\int{\sqrt{1-x^2}dx}\)
\(=\frac{1}{2}x^2\sin^{-1}{x}-\frac{1}{2}\sin^{-1}{x}+\frac{1}{2}\left(\frac{x\sqrt{1-x^2}}{2}+\frac{1}{2}\sin^{-1}{x}\right)+c\) ➜Note \(\because \int{\frac{1}{\sqrt{1-x^2}}dx}=\sin^{-1}{x}\), \(\int{\sqrt{a^2-x^2}dx}=\frac{x\sqrt{a^2-x^2}}{2}+\frac{a^2}{2}\sin^{-1}{\left(\frac{x}{a}\right)}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{2}x^2\sin^{-1}{x}+\frac{1}{4}x\sqrt{1-x^2}+\frac{1}{4}\sin^{-1}{x}-\frac{1}{2}\sin^{-1}{x}+c\)
\(=\frac{1}{2}x^2\sin^{-1}{x}+\frac{1}{4}x\sqrt{1-x^2}+\frac{1-2}{4}\sin^{-1}{x}+c\)
\(=\frac{1}{2}x^2\sin^{-1}{x}+\frac{1}{4}x\sqrt{1-x^2}+\frac{-1}{4}\sin^{-1}{x}+c\)
\(=\frac{1}{2}x^2\sin^{-1}{x}+\frac{1}{4}x\sqrt{1-x^2}-\frac{1}{4}\sin^{-1}{x}+c\)

\(\int{x \ cosec^{-1}{\frac{1}{x}}dx}\)
\(=\int{x\sin^{-1}{x}dx}\) ➜Note \(\because cosec^{-1}{\frac{1}{x}}=\sin^{-1}{x}\)
\(=\int{\sin^{-1}{x}.xdx}\)
\(=\sin^{-1}{x}\int{xdx}-\int{\left\{\frac{d}{dx}(\sin^{-1}{x})\int{xdx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\sin^{-1}{x}.\frac{x^2}{2}-\int{\frac{1}{\sqrt{1-x^2}}.\frac{x^2}{2}dx}\) ➜Note \(\because \int{xdx}=\frac{x^2}{2}, \frac{d}{dx}(\sin^{-1}{x})=\frac{1}{\sqrt{1-x^2}}\)
\(=\frac{1}{2}x^2\sin^{-1}{x}-\frac{1}{2}\int{\frac{x^2}{\sqrt{1-x^2}}dx}\)
\(=\frac{1}{2}x^2\sin^{-1}{x}-\frac{1}{2}\int{\frac{1-(1-x^2)}{\sqrt{1-x^2}}dx}\)
\(=\frac{1}{2}x^2\sin^{-1}{x}-\frac{1}{2}\int{\left(\frac{1}{\sqrt{1-x^2}}-\frac{1-x^2}{\sqrt{1-x^2}}\right)dx}\)
\(=\frac{1}{2}x^2\sin^{-1}{x}-\frac{1}{2}\int{\left(\frac{1}{\sqrt{1-x^2}}-\frac{\sqrt{1-x^2}.\sqrt{1-x^2}}{\sqrt{1-x^2}}\right)dx}\)
\(=\frac{1}{2}x^2\sin^{-1}{x}-\frac{1}{2}\int{\left(\frac{1}{\sqrt{1-x^2}}-\sqrt{1-x^2}\right)dx}\)
\(=\frac{1}{2}x^2\sin^{-1}{x}-\frac{1}{2}\int{\frac{1}{\sqrt{1-x^2}}dx}+\frac{1}{2}\int{\sqrt{1-x^2}dx}\)
\(=\frac{1}{2}x^2\sin^{-1}{x}-\frac{1}{2}\sin^{-1}{x}+\frac{1}{2}\left(\frac{x\sqrt{1-x^2}}{2}+\frac{1}{2}\sin^{-1}{x}\right)+c\) ➜Note \(\because \int{\frac{1}{\sqrt{1-x^2}}dx}=\sin^{-1}{x}\), \(\int{\sqrt{a^2-x^2}dx}=\frac{x\sqrt{a^2-x^2}}{2}+\frac{a^2}{2}\sin^{-1}{\left(\frac{x}{a}\right)}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{2}x^2\sin^{-1}{x}+\frac{1}{4}x\sqrt{1-x^2}+\frac{1}{4}\sin^{-1}{x}-\frac{1}{2}\sin^{-1}{x}+c\)
\(=\frac{1}{2}x^2\sin^{-1}{x}+\frac{1}{4}x\sqrt{1-x^2}+\frac{1-2}{4}\sin^{-1}{x}+c\)
\(=\frac{1}{2}x^2\sin^{-1}{x}+\frac{1}{4}x\sqrt{1-x^2}+\frac{-1}{4}\sin^{-1}{x}+c\)
\(=\frac{1}{2}x^2\sin^{-1}{x}+\frac{1}{4}x\sqrt{1-x^2}-\frac{1}{4}\sin^{-1}{x}+c\)

\(\int{mx\sin^{-1}{x}dx}\)
\(=m\int{\sin^{-1}{x}.xdx}\)
\(=m\left[\sin^{-1}{x}\int{xdx}-\int{\left\{\frac{d}{dx}(\sin^{-1}{x})\int{xdx}\right\}dx}\right]\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=m\left[\sin^{-1}{x}.\frac{x^2}{2}-\int{\frac{1}{\sqrt{1-x^2}}.\frac{x^2}{2}dx}\right]\) ➜Note \(\because \int{xdx}=\frac{x^2}{2}, \frac{d}{dx}(\sin^{-1}{x})=\frac{1}{\sqrt{1-x^2}}\)
\(=m\left[\frac{1}{2}x^2\sin^{-1}{x}-\frac{1}{2}\int{\frac{x^2}{\sqrt{1-x^2}}dx}\right]\)
\(=m\left[\frac{1}{2}x^2\sin^{-1}{x}-\frac{1}{2}\int{\frac{1-(1-x^2)}{\sqrt{1-x^2}}dx}\right]\)
\(=m\left[\frac{1}{2}x^2\sin^{-1}{x}-\frac{1}{2}\int{\left(\frac{1}{\sqrt{1-x^2}}-\frac{1-x^2}{\sqrt{1-x^2}}\right)dx}\right]\)
\(=m\left[\frac{1}{2}x^2\sin^{-1}{x}-\frac{1}{2}\int{\left(\frac{1}{\sqrt{1-x^2}}-\frac{\sqrt{1-x^2}.\sqrt{1-x^2}}{\sqrt{1-x^2}}\right)dx}\right]\)
\(=m\left[\frac{1}{2}x^2\sin^{-1}{x}-\frac{1}{2}\int{\left(\frac{1}{\sqrt{1-x^2}}-\sqrt{1-x^2}\right)dx}\right]\)
\(=m\left[\frac{1}{2}x^2\sin^{-1}{x}-\frac{1}{2}\int{\frac{1}{\sqrt{1-x^2}}dx}+\frac{1}{2}\int{\sqrt{1-x^2}dx}\right]\)
\(=m\left[\frac{1}{2}x^2\sin^{-1}{x}-\frac{1}{2}\sin^{-1}{x}+\frac{1}{2}\left(\frac{x\sqrt{1-x^2}}{2}+\frac{1}{2}\sin^{-1}{x}\right)\right]+c\) ➜Note \(\because \int{\frac{1}{\sqrt{1-x^2}}dx}=\sin^{-1}{x}\), \(\int{\sqrt{a^2-x^2}dx}=\frac{x\sqrt{a^2-x^2}}{2}+\frac{a^2}{2}\sin^{-1}{\left(\frac{x}{a}\right)}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=m\left(\frac{1}{2}x^2\sin^{-1}{x}+\frac{1}{4}x\sqrt{1-x^2}+\frac{1}{4}\sin^{-1}{x}-\frac{1}{2}\sin^{-1}{x}\right)+c\)
\(=m\left(\frac{1}{2}x^2\sin^{-1}{x}+\frac{1}{4}x\sqrt{1-x^2}+\frac{1-2}{4}\sin^{-1}{x}\right)+c\)
\(=m\left(\frac{1}{2}x^2\sin^{-1}{x}+\frac{1}{4}x\sqrt{1-x^2}+\frac{-1}{4}\sin^{-1}{x}\right)+c\)
\(=m\left(\frac{1}{2}x^2\sin^{-1}{x}+\frac{1}{4}x\sqrt{1-x^2}-\frac{1}{4}\sin^{-1}{x}\right)+c\)

\(\int{\tan^{-1}{x}dx}\)
\(=\int{\tan^{-1}{x}.1dx}\)
\(=\tan^{-1}{x}\int{1dx}-\int{\left\{\frac{d}{dx}(\tan^{-1}{x})\int{1dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\tan^{-1}{x}.x-\int{\frac{1}{1+x^2}.xdx}\) ➜Note \(\because \int{1dx}=x, \frac{d}{dx}(\tan^{-1}{x})=\frac{1}{1+x^2}\)
\(=x\tan^{-1}{x}-\int{\frac{x}{1+x^2}dx}\)
\(=x\tan^{-1}{x}-\frac{1}{2}\int{\frac{2x}{1+x^2}dx}\)
\(=x\tan^{-1}{x}-\frac{1}{2}\int{\frac{d(1+x^2)}{1+x^2}}\)
\(=x\tan^{-1}{x}-\frac{1}{2}\ln{|1+x^2|}+c\) ➜Note \(\because \int{\frac{dx}{x}}=\ln{|x|}\) এবং \(c\) যোগজীকরণ ধ্রুবক।

\(\int{\tan^{-1}{\frac{x}{5}}dx}\)
\(=\int{\tan^{-1}{\frac{x}{5}}.1dx}\)
\(=\tan^{-1}{\frac{x}{5}}\int{1dx}-\int{\left\{\frac{d}{dx}(\tan^{-1}{\frac{x}{5}})\int{1dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\tan^{-1}{\frac{x}{5}}.x-\int{\frac{1}{1+\left(\frac{x}{5}\right)^2}.\frac{d}{dx}\left(\frac{x}{5}\right).xdx}\) ➜Note \(\because \int{1dx}=x, \frac{d}{dx}(\tan^{-1}{x})=\frac{1}{1+x^2}\)
\(=x\tan^{-1}{\frac{x}{5}}-\int{\frac{1}{1+\frac{x^2}{25}}.\frac{1}{5}.xdx}\)
\(=x\tan^{-1}{\frac{x}{5}}-\frac{1}{5}\int{\frac{1}{\frac{25+x^2}{25}}.xdx}\)
\(=x\tan^{-1}{\frac{x}{5}}-\frac{1}{5}\int{\frac{25x}{25+x^2}dx}\)
\(=x\tan^{-1}{\frac{x}{5}}-\int{\frac{5x}{25+x^2}dx}\)
\(=x\tan^{-1}{\frac{x}{5}}-\frac{5}{2}\int{\frac{2x}{25+x^2}dx}\)
\(=x\tan^{-1}{\frac{x}{5}}-\frac{5}{2}\int{\frac{d(25+x^2)}{25+x^2}}\)
\(=x\tan^{-1}{\frac{x}{5}}-\frac{5}{2}\ln{|25+x^2|}+c\) ➜Note \(\because \int{\frac{dx}{x}}=\ln{|x|}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=x\tan^{-1}{\frac{x}{5}}-\frac{5}{2}\ln{|x^2+25|}+c\)

\(\int{e^{2x}\cos{x}dx}\)
ধরি,
\(I=\int{\cos{x}e^{2x}dx}\)
\(=\cos{x}\int{e^{2x}dx}-\int{\left\{\frac{d}{dx}(\cos{x})\int{e^{2x}dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\cos{x}.\frac{e^{2x}}{2}-\int{(-\sin{x}).\frac{e^{2x}}{2}dx}\) ➜Note \(\because \int{e^{ax}dx}=\frac{e^{ax}}{a}, \frac{d}{dx}(\cos{x})=-\sin{x}\)
\(=\frac{1}{2}e^{2x}\cos{x}+\frac{1}{2}\int{\sin{x}e^{2x}dx}\)
\(=\frac{1}{2}e^{2x}\cos{x}+\frac{1}{2}\left[\sin{x}\int{e^{2x}dx}-\int{\left\{\frac{d}{dx}(\sin{x})\int{e^{2x}dx}\right\}dx}\right]\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{1}{2}e^{2x}\cos{x}+\frac{1}{2}\left[\sin{x}.\frac{e^{2x}}{2}-\int{\cos{x}.\frac{e^{2x}}{2}dx}\right]\) ➜Note \(\because \int{e^{ax}dx}=\frac{e^{ax}}{a}, \frac{d}{dx}(\sin{x})=\cos{x}\)
\(=\frac{1}{2}e^{2x}\cos{x}+\frac{1}{2}\left[\frac{1}{2}e^{2x}\sin{x}-\frac{1}{2}\int{\cos{x}e^{2x}dx}\right]\)
\(=\frac{1}{2}e^{2x}\cos{x}+\frac{1}{4}e^{2x}\sin{x}-\frac{1}{4}\int{\cos{x}e^{2x}dx}\)
\(I=\frac{1}{2}e^{2x}\cos{x}+\frac{1}{4}e^{2x}\sin{x}-\frac{1}{4}I\) ➜Note \(\because \int{\cos{x}e^{2x}dx}=I\)
\(\Rightarrow I+\frac{1}{4}I=\frac{1}{2}e^{2x}\cos{x}+\frac{1}{4}e^{2x}\sin{x}\)
\(\Rightarrow \frac{4+1}{4}I=\frac{1}{2}e^{2x}\cos{x}+\frac{1}{4}e^{2x}\sin{x}\)
\(\Rightarrow \frac{5}{4}I=\frac{1}{2}e^{2x}\cos{x}+\frac{1}{4}e^{2x}\sin{x}\)
\(\Rightarrow I=\frac{4}{5}.\frac{1}{2}e^{2x}\cos{x}+\frac{4}{5}.\frac{1}{4}e^{2x}\sin{x}+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।
\(\Rightarrow \int{\cos{x}e^{2x}dx}=\frac{2}{5}e^{2x}\cos{x}+\frac{1}{5}e^{2x}\sin{x}+c\)
\(\therefore \int{e^{2x}\cos{x}dx}=\frac{1}{5}e^{2x}(2\cos{x}+\sin{x})+c\)
\(\int{e^{2x}\cos{x}dx}\)
\(=\frac{e^{2x}}{2^2+1^2}(2\cos{x}+1\sin{x})+c\) ➜Note \(\because \int{e^{ax}\cos{bx}dx}=\frac{e^{ax}}{a^2+b^2}(a\cos{bx}+b\sin{bx})\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{e^{2x}}{4+1}(2\cos{x}+\sin{x})+c\)
\(=\frac{e^{2x}}{5}(2\cos{x}+\sin{x})+c\)
\(=\frac{1}{5}e^{2x}(2\cos{x}+\sin{x})+c\)

\(\int{e^{2x}\sin{x}dx}\)
ধরি,
\(I=\int{\sin{x}e^{2x}dx}\)
\(=\sin{x}\int{e^{2x}dx}-\int{\left\{\frac{d}{dx}(\sin{x})\int{e^{2x}dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\sin{x}.\frac{e^{2x}}{2}-\int{\cos{x}.\frac{e^{2x}}{2}dx}\) ➜Note \(\because \int{e^{ax}dx}=\frac{e^{ax}}{a}, \frac{d}{dx}(\sin{x})=\cos{x}\)
\(=\frac{1}{2}e^{2x}\sin{x}-\frac{1}{2}\int{\cos{x}e^{2x}dx}\)
\(=\frac{1}{2}e^{2x}\sin{x}-\frac{1}{2}\left[\cos{x}\int{e^{2x}dx}-\int{\left\{\frac{d}{dx}(\cos{x})\int{e^{2x}dx}\right\}dx}\right]\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{1}{2}e^{2x}\sin{x}-\frac{1}{2}\left[\cos{x}.\frac{e^{2x}}{2}-\int{(-\sin{x}).\frac{e^{2x}}{2}dx}\right]\) ➜Note \(\because \int{e^{ax}dx}=\frac{e^{ax}}{a}, \frac{d}{dx}(\cos{x})=-\sin{x}\)
\(=\frac{1}{2}e^{2x}\sin{x}-\frac{1}{2}\left[\frac{1}{2}e^{2x}\cos{x}+\frac{1}{2}\int{\sin{x}e^{2x}dx}\right]\)
\(=\frac{1}{2}e^{2x}\sin{x}-\frac{1}{4}e^{2x}\cos{x}-\frac{1}{4}\int{\sin{x}e^{2x}dx}\)
\(I=\frac{1}{2}e^{2x}\sin{x}-\frac{1}{4}e^{2x}\cos{x}-\frac{1}{4}I\) ➜Note \(\because \int{\sin{x}e^{2x}dx}=I\)
\(\Rightarrow I+\frac{1}{4}I=\frac{1}{2}e^{2x}\sin{x}-\frac{1}{4}e^{2x}\cos{x}\)
\(\Rightarrow \frac{4+1}{4}I=\frac{1}{2}e^{2x}\sin{x}-\frac{1}{4}e^{2x}\cos{x}\)
\(\Rightarrow \frac{5}{4}I=\frac{1}{2}e^{2x}\sin{x}-\frac{1}{4}e^{2x}\cos{x}\)
\(\Rightarrow I=\frac{4}{5}.\frac{1}{2}e^{2x}\sin{x}-\frac{4}{5}.\frac{1}{4}e^{2x}\cos{x}+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।
\(\Rightarrow \int{\sin{x}e^{2x}dx}=\frac{2}{5}e^{2x}\sin{x}-\frac{1}{5}e^{2x}\cos{x}+c\)
\(\therefore \int{e^{2x}\sin{x}dx}=\frac{1}{5}e^{2x}(2\sin{x}-\cos{x})+c\)
\(\int{e^{2x}\sin{x}dx}\)
\(=\frac{e^{2x}}{2^2+1^2}(2\sin{x}-1\cos{x})+c\) ➜Note \(\because \int{e^{ax}\sin{bx}dx}=\frac{e^{ax}}{a^2+b^2}(a\sin{bx}-b\cos{bx})\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{e^{2x}}{4+1}(2\sin{x}-\cos{x})+c\)
\(=\frac{e^{2x}}{5}(2\sin{x}-\cos{x})+c\)
\(=\frac{1}{5}e^{2x}(2\sin{x}-\cos{x})+c\)

\(\int{e^{x}\sin{2x}dx}\)
ধরি,
\(I=\int{\sin{2x}e^{x}dx}\)
\(=\sin{2x}\int{e^{x}dx}-\int{\left\{\frac{d}{dx}(\sin{2x})\int{e^{x}dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\sin{2x}.e^{x}-\int{\cos{2x}.2.e^xdx}\) ➜Note \(\because \int{e^{x}dx}=e^x, \frac{d}{dx}(\sin{2x})=\cos{2x}.2\)
\(=e^{x}\sin{2x}-2\int{\cos{2x}e^{x}dx}\)
\(=e^{x}\sin{2x}-2\left[\cos{2x}\int{e^{x}dx}-\int{\left\{\frac{d}{dx}(\cos{2x})\int{e^{x}dx}\right\}dx}\right]\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=e^{x}\sin{2x}-2\left[\cos{2x}.e^x-\int{(-\sin{2x}).2.e^xdx}\right]\) ➜Note \(\because \int{e^{x}dx}=e^x, \frac{d}{dx}(\cos{2x})=-\sin{2x}.2\)
\(=e^{x}\sin{2x}-2\left[e^{x}\cos{2x}+2\int{\sin{2x}e^{x}dx}\right]\)
\(=e^{x}\sin{2x}-2e^{x}\cos{2x}-4\int{\sin{2x}e^{x}dx}\)
\(I=e^{x}\sin{2x}-2e^{x}\cos{2x}-4I\) ➜Note \(\because \int{\sin{2x}e^{x}dx}=I\)
\(\Rightarrow I+4I=e^{x}\sin{2x}-2e^{x}\cos{2x}\)
\(\Rightarrow 5I=e^{x}\sin{2x}-2e^{x}\cos{2x}\)
\(\Rightarrow I=\frac{1}{5}e^{x}\sin{2x}-\frac{1}{5}.2e^{x}\cos{2x}\)
\(\Rightarrow I=\frac{1}{5}e^{x}\sin{2x}-\frac{2}{5}e^{x}\cos{2x}+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।
\(\Rightarrow \int{\sin{2x}e^{x}dx}=\frac{1}{5}e^{x}\sin{2x}-\frac{2}{5}e^{x}\cos{2x}+c\)
\(\therefore \int{e^{x}\sin{2x}dx}=\frac{1}{5}e^{x}(\sin{2x}-2\cos{2x})+c\)
\(\int{e^{x}\sin{2x}dx}\)
\(=\frac{e^{x}}{1^2+2^2}(1\sin{2x}-2\cos{2x})+c\) ➜Note \(\because \int{e^{ax}\sin{bx}dx}=\frac{e^{ax}}{a^2+b^2}(a\sin{bx}-b\cos{bx})\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{e^{x}}{1+4}(\sin{2x}-2\cos{2x})+c\)
\(=\frac{e^{2x}}{5}(\sin{2x}-2\cos{2x})+c\)
\(=\frac{1}{5}e^{x}(\sin{2x}-2\cos{2x})+c\)

\(\int{e^{x}\cos{x}dx}\)
ধরি,
\(I=\int{\cos{x}e^{x}dx}\)
\(=\cos{x}\int{e^{x}dx}-\int{\left\{\frac{d}{dx}(\cos{x})\int{e^{x}dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\cos{x}.e^x-\int{(-\sin{x}).e^xdx}\) ➜Note \(\because \int{e^{x}dx}=e^x, \frac{d}{dx}(\cos{x})=-\sin{x}\)
\(=e^{x}\cos{x}+\int{\sin{x}e^{x}dx}\)
\(=e^{x}\cos{x}+\left[\sin{x}\int{e^{x}dx}-\int{\left\{\frac{d}{dx}(\sin{x})\int{e^{x}dx}\right\}dx}\right]\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=e^{x}\cos{x}+\left[\sin{x}.e^x-\int{\cos{x}.e^xdx}\right]\) ➜Note \(\because \int{e^{x}dx}=e^x, \frac{d}{dx}(\sin{x})=\cos{x}\)
\(=e^{x}\cos{x}+\left[e^{x}\sin{x}-\int{\cos{x}e^{x}dx}\right]\)
\(=e^{x}\cos{x}+e^{x}\sin{x}-\int{\cos{x}e^{x}dx}\)
\(I=e^{x}\cos{x}+e^{x}\sin{x}-I\) ➜Note \(\because \int{\cos{x}e^{2x}dx}=I\)
\(\Rightarrow I+I=e^{x}\cos{x}+e^{x}\sin{x}\)
\(\Rightarrow 2I=e^{x}(\cos{x}+\sin{x})\)
\(\Rightarrow I=\frac{1}{2}e^{x}(\cos{x}+\sin{x})+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।
\(\Rightarrow \int{\cos{x}e^{x}dx}=\frac{1}{2}e^{x}(\cos{x}+\sin{x})+c\)
\(\therefore \int{e^{x}\cos{x}dx}=\frac{1}{2}e^{x}(\cos{x}+\sin{x})+c\)
\(\int{e^{x}\cos{x}dx}\)
\(=\frac{e^{x}}{1^2+1^2}(1\cos{x}+1\sin{x})+c\) ➜Note \(\because \int{e^{ax}\cos{bx}dx}=\frac{e^{ax}}{a^2+b^2}(a\cos{bx}+b\sin{bx})\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{e^{x}}{1+1}(\cos{x}+\sin{x})+c\)
\(=\frac{e^{x}}{2}(\cos{x}+\sin{x})+c\)
\(=\frac{1}{2}e^{x}(\cos{x}+\sin{x})+c\)

\(\int{e^{x}\sin{x}dx}\)
ধরি,
\(I=\int{\sin{x}e^{x}dx}\)
\(=\sin{x}\int{e^{x}dx}-\int{\left\{\frac{d}{dx}(\sin{x})\int{e^{x}dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\sin{x}.e^{x}-\int{\cos{x}.e^xdx}\) ➜Note \(\because \int{e^{x}dx}=e^x, \frac{d}{dx}(\sin{x})=\cos{x}\)
\(=e^{x}\sin{x}-\int{\cos{x}e^{x}dx}\)
\(=e^{x}\sin{2x}-\left[\cos{x}\int{e^{x}dx}-\int{\left\{\frac{d}{dx}(\cos{x})\int{e^{x}dx}\right\}dx}\right]\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=e^{x}\sin{x}-\left[\cos{x}.e^x-\int{(-\sin{x}).e^xdx}\right]\) ➜Note \(\because \int{e^{x}dx}=e^x, \frac{d}{dx}(\cos{x})=-\sin{x}\)
\(=e^{x}\sin{x}-\left[e^{x}\cos{x}+\int{\sin{x}e^{x}dx}\right]\)
\(=e^{x}\sin{x}-e^{x}\cos{x}-\int{\sin{x}e^{x}dx}\)
\(I=e^{x}\sin{x}-e^{x}\cos{x}-I\) ➜Note \(\because \int{\sin{2x}e^{x}dx}=I\)
\(\Rightarrow I+I=e^{x}\sin{x}-e^{x}\cos{x}\)
\(\Rightarrow 2I=e^{x}\sin{x}-e^{x}\cos{x}\)
\(\Rightarrow I=\frac{1}{2}e^{x}\sin{x}-\frac{1}{2}e^{x}\cos{x}\)
\(\Rightarrow I=\frac{1}{2}e^{x}\sin{x}-\frac{1}{2}e^{x}\cos{x}+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।
\(\Rightarrow \int{\sin{x}e^{x}dx}=\frac{1}{2}e^{x}\sin{x}-\frac{1}{2}e^{x}\cos{x}+c\)
\(\therefore \int{e^{x}\sin{x}dx}=\frac{1}{2}e^{x}(\sin{x}-\cos{x})+c\)
\(\int{e^{x}\sin{x}dx}\)
\(=\frac{e^{x}}{1^2+1^2}(1\sin{x}-1\cos{x})+c\) ➜Note \(\because \int{e^{ax}\sin{bx}dx}=\frac{e^{ax}}{a^2+b^2}(a\sin{bx}-b\cos{bx})\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{e^{x}}{1+1}(\sin{x}-\cos{x})+c\)
\(=\frac{e^{x}}{2}(\sin{x}-\cos{x})+c\)
\(=\frac{1}{2}e^{x}(\sin{x}-\cos{x})+c\)

\(\int{e^{ax}\sin{bx}dx}\)
ধরি,
\(I=\int{e^{ax}\sin{bx}dx}\)
\(=\int{\sin{bx}.e^{ax}dx}\)
\(=\sin{bx}\int{e^{ax}dx}-\int{\left\{\frac{d}{dx}(\sin{bx})\int{e^{ax}dx}\right\}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\sin{bx}.\frac{e^{ax}}{a}-\int{\left\{\cos{bx}.\frac{d}{dx}(bx).\frac{e^{ax}}{a}\right\}dx}\) ➜Note \(\because \int{e^{ax}dx}=\frac{e^{ax}}{a}, \frac{d}{dx}(\sin{bx})=b\cos{bx}\)
\(=\frac{1}{a}e^{ax}\sin{bx}-\int{\left\{\cos{bx}.b.\frac{e^{ax}}{a}\right\}dx}\)
\(=\frac{1}{a}e^{ax}\sin{bx}-\frac{b}{a}\int{\cos{bx}.e^{ax}dx}\)
\(=\frac{1}{a}e^{ax}\sin{bx}-\frac{b}{a}\left[\cos{bx}\int{e^{ax}dx}-\int{\left\{\frac{d}{dx}(\cos{bx})\int{e^{ax}dx}\right\}dx}\right]\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{1}{a}e^{ax}\sin{bx}-\frac{b}{a}\left[\cos{bx}.\frac{e^{ax}}{a}-\int{(-\sin{bx}).\frac{d}{dx}(bx).\frac{e^{ax}}{a}dx}\right]\) ➜Note \(\because \int{e^{ax}dx}=\frac{e^{ax}}{a}, \frac{d}{dx}(\cos{bx})=-b\sin{bx}\)
\(=\frac{1}{a}e^{ax}\sin{bx}-\frac{b}{a}\left[\frac{1}{a}e^{ax}\cos{bx}+\int{\sin{bx}.b.\frac{e^{ax}}{a}dx}\right]\)
\(=\frac{1}{a}e^{ax}\sin{bx}-\frac{b}{a}\left[\frac{1}{a}e^{ax}\cos{bx}+\frac{b}{a}\int{\sin{bx}.e^{ax}dx}\right]\)
\(=\frac{1}{a}e^{ax}\sin{bx}-\frac{b}{a^2}e^{ax}\cos{bx}-\frac{b^2}{a^2}\int{\sin{bx}.e^{ax}dx}\)
\(I=\frac{1}{a}e^{ax}\sin{bx}-\frac{b}{a^2}e^{ax}\cos{bx}-\frac{b^2}{a^2}I\) ➜Note \(\because \int{\sin{bx}.e^{ax}dx}=I\)
\(\Rightarrow I+\frac{b^2}{a^2}I=\frac{1}{a}e^{ax}\sin{bx}-\frac{b}{a^2}e^{ax}\cos{bx}\)
\(\Rightarrow I\left(1+\frac{b^2}{a^2}\right)=\frac{1}{a^2}e^{ax}(a\sin{bx}-b\cos{bx})\)
\(\Rightarrow I\left(\frac{a^2+b^2}{a^2}\right)=\frac{1}{a^2}e^{ax}(a\sin{bx}-b\cos{bx})\)
\(\Rightarrow I=\frac{a^2}{a^2+b^2}\times{\frac{1}{a^2}}e^{ax}(a\sin{bx}-b\cos{bx})+c\) ➜Note\(c\) যোগজীকরণ ধ্রুবক।
\(\Rightarrow I=\frac{e^{ax}}{a^2+b^2}(a\sin{bx}-b\cos{bx})+c\)
\(\therefore \int{e^{ax}\sin{bx}dx}=\frac{e^{ax}}{a^2+b^2}(a\sin{bx}-b\cos{bx})+c\)

\(\int{x\cos^2{x}dx}\)
\(=\frac{1}{2}\int{x.2\cos^2{x}dx}\)
\(=\frac{1}{2}\int{x(1+\cos{2x})dx}\) ➜Note \(\because 2\cos^2{A}=1+\cos{2A}\)
\(=\frac{1}{2}\int{(x+x\cos{2x})dx}\)
\(=\frac{1}{2}\int{xdx}+\frac{1}{2}\int{x\cos{2x}dx}\)
\(=\frac{1}{2}\frac{x^2}{2}+\frac{1}{2}\left[x\int{\cos{2x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\cos{2x}dx}\right\}dx}\right]\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{1}{4}x^2+\frac{1}{2}\left[x\frac{\sin{2x}}{2}-\int{1.\frac{\sin{2x}}{2}dx}\right]\) ➜Note \(\because \frac{d}{dx}(x)=1, \int{\cos{ax}dx}=\frac{\sin{ax}}{a}\)
\(=\frac{1}{4}x^2+\frac{1}{4}x\sin{2x}-\frac{1}{4}\int{\sin{2x}dx}\)
\(=\frac{1}{4}x^2+\frac{1}{4}x\sin{2x}-\frac{1}{4}\times{-\frac{\cos{2x}}{2}}+c\) ➜Note\(\because \int{\sin{ax}dx}=-\frac{\cos{ax}}{a}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{4}x^2+\frac{1}{4}x\sin{2x}+\frac{1}{8}\cos{2x}+c\)
\(=\frac{1}{8}(2x^2+2x\sin{2x}+\cos{2x})+c\)

\(\int{x\sin^2{x}dx}\)
\(=\frac{1}{2}\int{x.2\sin^2{x}dx}\)
\(=\frac{1}{2}\int{x(1-\cos{2x})dx}\) ➜Note \(\because 2\sin^2{A}=1-\cos{2A}\)
\(=\frac{1}{2}\int{(x-x\cos{2x})dx}\)
\(=\frac{1}{2}\int{xdx}-\frac{1}{2}\int{x\cos{2x}dx}\)
\(=\frac{1}{2}\frac{x^2}{2}-\frac{1}{2}\left[x\int{\cos{2x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\cos{2x}dx}\right\}dx}\right]\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{1}{4}x^2-\frac{1}{2}\left[x\frac{\sin{2x}}{2}-\int{1.\frac{\sin{2x}}{2}dx}\right]\) ➜Note \(\because \frac{d}{dx}(x)=1, \int{\cos{ax}dx}=\frac{\sin{ax}}{a}\)
\(=\frac{1}{4}x^2-\frac{1}{4}x\sin{2x}+\frac{1}{4}\int{\sin{2x}dx}\)
\(=\frac{1}{4}x^2-\frac{1}{4}x\sin{2x}+\frac{1}{4}\times{-\frac{\cos{2x}}{2}}+c\) ➜Note\(\because \int{\sin{ax}dx}=-\frac{\cos{ax}}{a}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{4}x^2-\frac{1}{4}x\sin{2x}-\frac{1}{8}\cos{2x}+c\)
\(=\frac{1}{8}(2x^2-2x\sin{2x}-\cos{2x})+c\)

\(\int{x\sin{x}\cos{x}dx}\)
\(=\frac{1}{2}\int{x.2\sin{x}\cos{x}dx}\)
\(=\frac{1}{2}\int{x\sin{2x}dx}\) ➜Note \(\because 2\sin{A}\cos{A}=\sin{2A}\)
\(=\frac{1}{2}\left[x\int{\sin{2x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\sin{2x}dx}\right\}dx}\right]\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{1}{2}\left[x\left(-\frac{\cos{2x}}{2}\right)-\int{1.\left(-\frac{\cos{2x}}{2}\right)dx}\right]\) ➜Note \(\because \int{\sin{ax}dx}=-\frac{\cos{ax}}{a}\)
\(=\frac{1}{2}\left[-\frac{1}{2}x\cos{2x}+\frac{1}{2}\int{\cos{2x}dx}\right]\)
\(=-\frac{1}{4}x\cos{2x}+\frac{1}{4}\int{\cos{2x}dx}\)
\(=-\frac{1}{4}x\cos{2x}+\frac{1}{4}\times{\frac{\sin{2x}}{2}}+c\) ➜Note \(\because \int{\cos{ax}dx}=\frac{\sin{ax}}{a}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=-\frac{1}{4}x\cos{2x}+\frac{1}{8}\sin{2x}+c\)
\(=\frac{1}{8}\sin{2x}-\frac{1}{4}x\cos{2x}+c\)
\(=\frac{1}{8}(\sin{2x}-2x\cos{2x})+c\)

\(\int{x\sin^3{x}dx}\)
\(=\frac{1}{4}\int{x.4\sin^3{x}dx}\)
\(=\frac{1}{4}\int{x(3\sin{x}-\sin{3x})dx}\) ➜Note \(\because 4\sin^3{A}=3\sin{A}-\sin{3A}\)
\(=\frac{3}{4}\int{x\sin{x}dx}-\frac{1}{4}\int{x\sin{3x}dx}\)
\(=\frac{3}{4}\left[x\int{\sin{x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\sin{x}dx}\right\}dx}\right]-\frac{1}{4}\left[x\int{\sin{3x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\sin{3x}dx}\right\}dx}\right]\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{3}{4}\left[-x\cos{x}-\int{1\times{-\cos{x}}dx}\right]-\frac{1}{4}\left[x\times{-\frac{\cos{3x}}{3}}-\int{1\times{-\frac{\cos{3x}}{3}}dx}\right]\) ➜Note \(\because \int{\sin{x}dx}=-\cos{x}, \frac{d}{dx}(x)=1\), \(\int{\sin{3x}dx}=-\frac{\cos{3x}}{3}\)
\(=\frac{3}{4}\left[-x\cos{x}+\int{\cos{x}dx}\right]-\frac{1}{4}\left[-\frac{1}{3}x\cos{3x}+\frac{1}{3}\int{\cos{3x}dx}\right]\)
\(=-\frac{3}{4}x\cos{x}+\frac{3}{4}\int{\cos{x}dx}+\frac{1}{12}x\cos{3x}-\frac{1}{12}\int{\cos{3x}dx}\)
\(=-\frac{3}{4}x\cos{x}+\frac{3}{4}\sin{x}+\frac{1}{12}x\cos{3x}-\frac{1}{12}\frac{\sin{3x}}{3}+c\) ➜Note \(\because \int{\cos{3x}dx}=\frac{\sin{3x}}{3}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{3}{4}\sin{x}+\frac{1}{12}x\cos{3x}-\frac{3}{4}x\cos{x}-\frac{1}{36}\sin{3x}+c\)
\(=\frac{1}{36}(27\sin{x}+3x\cos{3x}-27x\cos{x}-\sin{3x})+c\)

\(\int{x\sin^2{\left(\frac{x}{2}\right)}dx}\)
\(=\frac{1}{2}\int{x.2\sin^2{\left(\frac{x}{2}\right)}dx}\)
\(=\frac{1}{2}\int{x(1-\cos{x})dx}\) ➜Note \(\because 2\sin^2{\left(\frac{A}{2}\right)}=1-\cos{A}\)
\(=\frac{1}{2}\int{(x-x\cos{x})dx}\)
\(=\frac{1}{2}\int{xdx}-\frac{1}{2}\int{x\cos{x}dx}\)
\(=\frac{1}{2}\frac{x^2}{2}-\frac{1}{2}\left[x\int{\cos{x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\cos{x}dx}\right\}dx}\right]\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{1}{4}x^2-\frac{1}{2}\left[x\sin{x}-\int{1.\sin{x}dx}\right]\) ➜Note \(\because \frac{d}{dx}(x)=1, \int{\cos{x}dx}=\sin{x}\)
\(=\frac{1}{4}x^2-\frac{1}{2}x\sin{x}+\frac{1}{2}\int{\sin{x}dx}\)
\(=\frac{1}{4}x^2-\frac{1}{2}x\sin{x}+\frac{1}{2}\times{-\cos{x}}+c\) ➜Note\(\because \int{\sin{x}dx}=-\cos{x}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{4}x^2-\frac{1}{2}x\sin{2x}-\frac{1}{2}\cos{x}+c\)
\(=\frac{1}{4}(x^2-2x\sin{2x}-2\cos{2x})+c\)

\(\int{x\cos{2x}\cos{3x}dx}\)
\(=\frac{1}{2}\int{x.2\cos{3x}\cos{2x}dx}\)
\(=\frac{1}{2}\int{x\{\cos{(3x-2x)}+\cos{(3x+2x)}\}dx}\) ➜Note \(\because 2\cos{A}\cos{B}=\cos{(A-B)}+\cos{(A+B)}\)
\(=\frac{1}{2}\int{x(\cos{x}+\cos{5x})dx}\)
\(=\frac{1}{2}\int{x\cos{x}dx}+\frac{1}{2}\int{x\cos{5x}dx}\)
\(=\frac{1}{2}\left[x\int{\cos{x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\cos{x}dx}\right\}dx}\right]+\frac{1}{2}\left[x\int{\cos{5x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\cos{5x}dx}\right\}dx}\right]\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{1}{2}\left[x\sin{x}-\int{1\times{\sin{x}}dx}\right]+\frac{1}{2}\left[x\times{\frac{\sin{5x}}{5}}-\int{1\times{\frac{\sin{5x}}{5}}dx}\right]\) ➜Note \(\because \int{\cos{x}dx}=\sin{x}, \frac{d}{dx}(x)=1\), \(\int{\cos{ax}dx}=\frac{\sin{ax}}{a}\)
\(=\frac{1}{2}\left[x\sin{x}-\int{\sin{x}dx}\right]+\frac{1}{2}\left[\frac{1}{5}x\sin{5x}-\frac{1}{5}\int{\sin{5x}dx}\right]\)
\(=\frac{1}{2}x\sin{x}-\frac{1}{2}\int{\sin{x}dx}+\frac{1}{10}x\sin{5x}-\frac{1}{10}\int{\sin{5x}dx}\)
\(=\frac{1}{2}x\sin{x}-\frac{1}{2}\times{-\cos{x}}+\frac{1}{10}x\sin{5x}-\frac{1}{10}\times{-\frac{\cos{5x}}{5}}+c\) ➜Note\(\because \int{\sin{x}dx}=-\cos{x}, \int{\sin{ax}dx}=-\frac{\cos{ax}}{a}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{2}x\sin{x}+\frac{1}{2}\cos{x}+\frac{1}{10}x\sin{5x}+\frac{1}{50}\cos{5x}+c\)
\(=\frac{1}{50}(25x\sin{x}+25\cos{x}+5x\sin{5x}+\cos{5x})+c\)

\(\int{x\sin{x}\sin{2x}dx}\)
\(=\frac{1}{2}\int{x.2\sin{2x}\sin{x}dx}\)
\(=\frac{1}{2}\int{x\{\cos{(2x-x)}-\cos{(2x+x)}\}dx}\) ➜Note \(\because 2\sin{A}\sin{B}=\cos{(A-B)}-\cos{(A+B)}\)
\(=\frac{1}{2}\int{x(\cos{x}-\cos{3x})dx}\)
\(=\frac{1}{2}\int{x\cos{x}dx}-\frac{1}{2}\int{x\cos{3x}dx}\)
\(=\frac{1}{2}\left[x\int{\cos{x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\cos{x}dx}\right\}dx}\right]-\frac{1}{2}\left[x\int{\cos{3x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\cos{3x}dx}\right\}dx}\right]\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{1}{2}\left[x\sin{x}-\int{1\times{\sin{x}}dx}\right]-\frac{1}{2}\left[x\times{\frac{\sin{3x}}{3}}-\int{1\times{\frac{\sin{3x}}{3}}dx}\right]\) ➜Note \(\because \int{\cos{x}dx}=\sin{x}, \frac{d}{dx}(x)=1\), \(\int{\cos{ax}dx}=\frac{\sin{ax}}{a}\)
\(=\frac{1}{2}\left[x\sin{x}-\int{\sin{x}dx}\right]-\frac{1}{2}\left[\frac{1}{3}x\sin{3x}-\frac{1}{3}\int{\sin{3x}dx}\right]\)
\(=\frac{1}{2}x\sin{x}-\frac{1}{2}\int{\sin{x}dx}-\frac{1}{6}x\sin{3x}+\frac{1}{6}\int{\sin{3x}dx}\)
\(=\frac{1}{2}x\sin{x}-\frac{1}{2}\times{-\cos{x}}-\frac{1}{6}x\sin{5x}+\frac{1}{6}\times{-\frac{\cos{3x}}{3}}+c\) ➜Note\(\because \int{\sin{x}dx}=-\cos{x}, \int{\sin{ax}dx}=-\frac{\cos{ax}}{a}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{2}x\sin{x}+\frac{1}{2}\cos{x}-\frac{1}{6}x\sin{5x}-\frac{1}{18}\cos{3x}+c\)
\(=\frac{1}{18}(9x\sin{x}+9\cos{x}-3x\sin{3x}-\cos{3x})+c\)

\(\int{x^2\cos^2{\left(\frac{x}{2}\right)}dx}\)
\(=\frac{1}{2}\int{x^2.2\cos^2{\left(\frac{x}{2}\right)}dx}\)
\(=\frac{1}{2}\int{x^2(1+\cos{x})dx}\) ➜Note \(\because 2\cos^2{\left(\frac{A}{2}\right)}=1+\cos{A}\)
\(=\frac{1}{2}\int{(x^2+x^2\cos{x})dx}\)
\(=\frac{1}{2}\int{x^2dx}+\frac{1}{2}\int{x^2\cos{x}dx}\)
\(=\frac{1}{2}\frac{x^3}{3}+\frac{1}{2}\left[x^2\int{\cos{x}dx}-\int{\left\{\frac{d}{dx}(x^2)\int{\cos{x}dx}\right\}dx}\right]\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{1}{6}x^3+\frac{1}{2}\left[x^2\sin{x}-\int{2x.\sin{x}dx}\right]\) ➜Note \(\because \frac{d}{dx}(x^2)=2x, \int{\cos{x}dx}=\sin{x}\)
\(=\frac{1}{6}x^3+\frac{1}{2}x^2\sin{x}-\int{x\sin{x}dx}\)
\(=\frac{1}{6}x^3+\frac{1}{2}x^2\sin{x}-\left[x\int{\sin{x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\sin{x}dx}\right\}dx}\right]\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{1}{6}x^3+\frac{1}{2}x^2\sin{x}-\left[-x\cos{x}-\int{1.(-\cos{x})dx}\right]\) ➜Note \(\because \frac{d}{dx}(x)=1, \int{\sin{x}dx}=-\cos{x}\)
\(=\frac{1}{6}x^3+\frac{1}{2}x^2\sin{x}-\left[-x\cos{x}+\int{\cos{x})dx}\right]\)
\(=\frac{1}{6}x^3+\frac{1}{2}x^2\sin{x}+x\cos{x}-\int{\cos{x})dx}\)
\(=\frac{1}{6}x^3+\frac{1}{2}x^2\sin{x}+x\cos{x}-\sin{x}+c\) ➜Note\(\because \int{\cos{x}dx}=\sin{x}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{6}(x^3+3x^2\sin{x}+6x\cos{x}-6\sin{x})+c\)

\(\int{x\tan^2{x}dx}\)
\(=\int{x(\sec^2{x}-1)dx}\) ➜Note \(\because \tan^2{A}=\sec^2{A}-1\)
\(=\int{(x\sec^2{x}-x)dx}\)
\(=\int{x\sec^2{x}dx}-\int{xdx}\)
\(=x\int{\sec^2{x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\sec^2{x}dx}\right\}dx}-\int{xdx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=x\tan{x}-\int{1.\tan{x}dx}-\frac{x^2}{2}\) ➜Note \(\because \int{\sec^2{x}dx}=\tan{x}, \frac{d}{dx}(x)=1\), \(\int{xdx}=\frac{x^2}{2}\)
\(=x\tan{x}-\int{\tan{x}dx}-\frac{x^2}{2}\)
\(=x\tan{x}+\ln{|\cos{x}|}-\frac{x^2}{2}+c\) ➜Note\(\because \int{\tan{x}dx}=-\ln{|\cos{x}|}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=x\tan{x}+\ln{|\cos{x}|}-\frac{1}{2}x^2+c\)

\(\int{x\sec^2{x}dx}\)
\(=x\int{\sec^2{x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\sec^2{x}dx}\right\}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=x\tan{x}-\int{1.\tan{x}dx}\) ➜Note \(\because \int{\sec^2{x}dx}=\tan{x}, \frac{d}{dx}(x)=1\)
\(=x\tan{x}-\int{\tan{x}dx}\)
\(=x\tan{x}+\ln{|\cos{x}|}+c\) ➜Note\(\because \int{\tan{x}dx}=-\ln{|\cos{x}|}\) এবং \(c\) যোগজীকরণ ধ্রুবক।

\(\int{x\tan^{-1}{(x^2)}dx}\)
\(=\int{\tan^{-1}{(x^2)}.xdx}\)
\(=\int{\tan^{-1}{t}.\frac{1}{2}dt}\)
\(=\frac{1}{2}\int{\tan^{-1}{t}.1dt}\)
\(=\frac{1}{2}\left[\tan^{-1}{t}\int{1dt}-\int{\left\{\frac{d}{dt}(\tan^{-1}{t})\int{1dt}\right\}dt}\right]\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{1}{2}\left[\tan^{-1}{t}.t-\int{\frac{1}{1+t^2}.tdt}\right]\) ➜Note \(\because \int{1dx}=x, \frac{d}{dx}(\tan^{-1}{x})=\frac{1}{1+x^2}\)
\(=\frac{1}{2}t\tan^{-1}{t}-\frac{1}{2}\int{\frac{t}{1+t^2}dt}\)
\(=\frac{1}{2}t\tan^{-1}{t}-\frac{1}{4}\int{\frac{2t}{1+t^2}dt}\)
\(=\frac{1}{2}t\tan^{-1}{t}-\frac{1}{4}\int{\frac{d(1+t^2)}{1+t^2}}\)
\(=\frac{1}{2}t\tan^{-1}{t}-\frac{1}{4}\ln{|1+t^2|}+c\) ➜Note\(\because \int{\frac{d(x)}{x}}=\ln{|x|}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{2}x^2\tan^{-1}{x^2}-\frac{1}{4}\ln{|1+x^4|}+c\) ➜Note\(\because t=x^2\)
\(=\frac{1}{4}(2x^2\tan^{-1}{x^2}-\ln{|1+x^4|})+c\) \(\int{x\tan^{-1}{(x^2)}dx}\)
\(=\int{\tan^{-1}{(x^2)}.xdx}\)
\(=\tan^{-1}{(x^2)}\int{xdx}-\int{\left\{\frac{d}{dt}(\tan^{-1}{(x^2)})\int{xdx}\right\}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\tan^{-1}{(x^2)}.\frac{x^2}{2}-\int{\frac{1}{1+x^4}.\frac{d}{dx}(x^2).\frac{x^2}{2}dx}\) ➜Note \(\because \int{xdx}=\frac{x^2}{2}, \frac{d}{dx}(\tan^{-1}{x})=\frac{1}{1+x^2}\)
\(=\frac{1}{2}x^2\tan^{-1}{(x^2)}-\int{\frac{1}{1+x^4}.2x.\frac{x^2}{2}dx}\) ➜Note \(\because \frac{d}{dx}(x^2)=2x\)
\(=\frac{1}{2}x^2\tan^{-1}{(x^2)}-\int{\frac{x^3}{1+x^4}dx}\)
\(=\frac{1}{2}x^2\tan^{-1}{(x^2)}-\frac{1}{4}\int{\frac{4x^3}{1+x^4}dx}\)
\(=\frac{1}{2}x^2\tan^{-1}{(x^2)}-\frac{1}{4}\int{\frac{d(1+x^4)}{1+x^4}}\)
\(=\frac{1}{2}x^2\tan^{-1}{(x^2)}-\frac{1}{4}\ln{|1+x^4|}+c\) ➜Note\(\because \int{\frac{d(x)}{x}}=\ln{|x|}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{4}(2x^2\tan^{-1}{x^2}-\ln{|1+x^4|})+c\)

\(\int{x\sin^{-1}{(x^2)}dx}\)
\(=\int{\sin^{-1}{(x^2)}.xdx}\)
\(=\int{\sin^{-1}{t}.\frac{1}{2}dt}\)
\(=\frac{1}{2}\int{\sin^{-1}{t}.1dt}\)
\(=\frac{1}{2}\left[\sin^{-1}{t}\int{1dt}-\int{\left\{\frac{d}{dt}(\sin^{-1}{t})\int{1dt}\right\}dt}\right]\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{1}{2}\left[\sin^{-1}{t}.t-\int{\frac{1}{\sqrt{1-t^2}}.tdt}\right]\) ➜Note \(\because \int{1dx}=x, \frac{d}{dx}(\sin^{-1}{x})=\frac{1}{\sqrt{1-t^2}}\)
\(=\frac{1}{2}t\sin^{-1}{t}-\frac{1}{2}\int{\frac{t}{\sqrt{1-t^2}}dt}\)
\(=\frac{1}{2}t\sin^{-1}{t}+\frac{1}{4}\int{\frac{-2t}{\sqrt{1-t^2}}dt}\)
\(=\frac{1}{2}t\sin^{-1}{t}+\frac{1}{4}\int{\frac{1}{\sqrt{1-t^2}}\times{-2tdt}}\)
\(=\frac{1}{2}t\sin^{-1}{t}+\frac{1}{4}\int{\frac{1}{\sqrt{z}}dz}\)
\(=\frac{1}{2}t\sin^{-1}{t}+\frac{1}{4}\int{\frac{1}{z^{\frac{1}{2}}}dz}\)
\(=\frac{1}{2}t\sin^{-1}{t}+\frac{1}{4}\int{z^{-\frac{1}{2}}dz}\)
\(=\frac{1}{2}t\sin^{-1}{t}+\frac{1}{4}\frac{z^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}+c\) ➜Note\(\because \int{x^ndx}=\frac{x^{n+1}}{n+1}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{2}t\sin^{-1}{t}+\frac{1}{4}\frac{z^{\frac{-1+2}{2}}}{\frac{-1+2}{2}}+c\)
\(=\frac{1}{2}t\sin^{-1}{t}+\frac{1}{4}\frac{z^{\frac{1}{2}}}{\frac{1}{2}}+c\)
\(=\frac{1}{2}t\sin^{-1}{t}+\frac{1}{2}z^{\frac{1}{2}}+c\)
\(=\frac{1}{2}t\sin^{-1}{t}+\frac{1}{2}\sqrt{z}+c\)
\(=\frac{1}{2}t\sin^{-1}{t}+\frac{1}{2}\sqrt{1-t^2}+c\) ➜Note\(\because z=1-t^2\)
\(=\frac{1}{2}x^2\sin^{-1}{(x^2)}+\frac{1}{2}\sqrt{1-x^4}+c\) ➜Note\(\because t=x^2\)
\(=\frac{1}{2}(x^2\sin^{-1}{(x^2)}+\sqrt{1-x^4})+c\) \(\int{x\sin^{-1}{(x^2)}dx}\)
\(=\int{\sin^{-1}{(x^2)}.xdx}\)
\(=\sin^{-1}{(x^2)}\int{xdx}-\int{\left\{\frac{d}{dt}(\sin^{-1}{(x^2)})\int{xdx}\right\}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\sin^{-1}{(x^2)}.\frac{x^2}{2}-\int{\frac{1}{\sqrt{1-x^4}}.\frac{d}{dx}(x^2).\frac{x^2}{2}dx}\) ➜Note \(\because \int{xdx}=\frac{x^2}{2}, \frac{d}{dx}(\sin^{-1}{x})=\frac{1}{\sqrt{1-x^2}}\)
\(=\frac{1}{2}x^2\sin^{-1}{(x^2)}-\int{\frac{1}{\sqrt{1-x^4}}.2x.\frac{x^2}{2}dx}\) ➜Note \(\because \frac{d}{dx}(x^2)=2x\)
\(=\frac{1}{2}x^2\sin^{-1}{(x^2)}-\int{\frac{x^3}{\sqrt{1-x^4}}dx}\)
\(=\frac{1}{2}x^2\sin^{-1}{(x^2)}+\frac{1}{4}\int{\frac{-4x^3}{\sqrt{1-x^4}}dx}\)
\(=\frac{1}{2}x^2\sin^{-1}{(x^2)}+\frac{1}{4}\int{\frac{1}{\sqrt{1-x^4}}\times{-4x^3dx}}\)
\(=\frac{1}{2}x^2\sin^{-1}{(x^2)}+\frac{1}{4}\int{\frac{1}{\sqrt{t}}dt}\)
\(=\frac{1}{2}x^2\sin^{-1}{(x^2)}+\frac{1}{4}\int{\frac{1}{t^{\frac{1}{2}}}dt}\)
\(=\frac{1}{2}x^2\sin^{-1}{(x^2)}+\frac{1}{4}\int{t^{-\frac{1}{2}}dt}\)
\(=\frac{1}{2}x^2\sin^{-1}{(x^2)}+\frac{1}{4}\frac{t^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}+c\) ➜Note\(\because \int{x^ndx}=\frac{x^{n+1}}{n+1}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{2}x^2\sin^{-1}{(x^2)}+\frac{1}{4}\frac{t^{\frac{-1+2}{2}}}{\frac{-1+2}{2}}+c\)
\(=\frac{1}{2}x^2\sin^{-1}{(x^2)}+\frac{1}{4}\frac{t^{\frac{1}{2}}}{\frac{1}{2}}+c\)
\(=\frac{1}{2}x^2\sin^{-1}{(x^2)}+\frac{1}{2}t^{\frac{1}{2}}+c\)
\(=\frac{1}{2}x^2\sin^{-1}{(x^2)}+\frac{1}{2}\sqrt{t}+c\)
\(=\frac{1}{2}x^2\sin^{-1}{(x^2)}+\frac{1}{2}\sqrt{1-x^4}+c\) ➜Note \(\because t=1-x^4\)
\(=\frac{1}{2}(x^2\sin^{-1}{(x^2)}+\sqrt{1-x^4})+c\)

\(\int{\cos^{-1}{\left(\frac{1-x^2}{1+x^2}\right)}dx}\)
\(=\int{\cos^{-1}{\left(\frac{1-\tan^2{\theta}}{1+\tan^2{\theta}}\right)}dx}\)
\(=\int{\cos^{-1}{\left(\cos{2\theta}\right)}dx}\) ➜Note \(\because \frac{1-\tan^2{A}}{1+\tan^2{A}}=\cos{2A}\)
\(=\int{2\theta.dx}\)
\(=2\int{\theta.dx}\)
\(=2\int{\tan^{-1}{x}dx}\) ➜Note \(\because \theta=\tan^{-1}{x}\)
\(=2\int{\tan^{-1}{x}.1dx}\)
\(=2\tan^{-1}{x}\int{1dx}-2\int{\left\{\frac{d}{dx}(\tan^{-1}{x})\int{1dx}\right\}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=2\tan^{-1}{x}.x-2\int{\left\{\frac{1}{1+x^2}.x\right\}dx}\) ➜Note \(\because \int{1dx}=x, \frac{d}{dx}(\tan^{-1}{x})=\frac{1}{1+x^2}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=2x\tan^{-1}{x}-2\int{\frac{1}{1+x^2}xdx}\)
\(=2x\tan^{-1}{x}-2\int{\frac{1}{t}\times{\frac{1}{2}}dt}\)
\(=2x\tan^{-1}{x}-\int{\frac{1}{t}dt}\)
\(=2x\tan^{-1}{x}-\ln{\left|t\right|}+c\) ➜Note \(\because \int{\frac{1}{x}dx}=\ln{|x|}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=2x\tan^{-1}{x}-\ln{|(1+x^2)|}+c\)

\(\int{\sin^{-1}{\left(\frac{2x}{1+x^2}\right)}dx}\)
\(=\int{\sin^{-1}{\left(\frac{2\tan{\theta}}{1+\tan^2{\theta}}\right)}dx}\)
\(=\int{\sin^{-1}{\left(\sin{2\theta}\right)}dx}\) ➜Note \(\because \frac{2\tan{A}}{1+\tan^2{A}}=\sin{2A}\)
\(=\int{2\theta.dx}\)
\(=2\int{\theta.dx}\)
\(=2\int{\tan^{-1}{x}dx}\) ➜Note \(\because \theta=\tan^{-1}{x}\)
\(=2\int{\tan^{-1}{x}.1dx}\)
\(=2\tan^{-1}{x}\int{1dx}-2\int{\left\{\frac{d}{dx}(\tan^{-1}{x})\int{1dx}\right\}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=2\tan^{-1}{x}.x-2\int{\left\{\frac{1}{1+x^2}.x\right\}dx}\) ➜Note \(\because \int{1dx}=x, \frac{d}{dx}(\tan^{-1}{x})=\frac{1}{1+x^2}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=2x\tan^{-1}{x}-2\int{\frac{1}{1+x^2}xdx}\)
\(=2x\tan^{-1}{x}-2\int{\frac{1}{t}\times{\frac{1}{2}}dt}\)
\(=2x\tan^{-1}{x}-\int{\frac{1}{t}dt}\)
\(=2x\tan^{-1}{x}-\ln{\left|t\right|}+c\) ➜Note \(\because \int{\frac{1}{x}dx}=\ln{|x|}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=2x\tan^{-1}{x}-\ln{|(1+x^2)|}+c\)

\(\int{\tan^{-1}{\left(\frac{2x}{1-x^2}\right)}dx}\)
\(=\int{\tan^{-1}{\left(\frac{2\tan{\theta}}{1-\tan^2{\theta}}\right)}dx}\)
\(=\int{\tan^{-1}{\left(\tan{2\theta}\right)}dx}\) ➜Note \(\because \frac{2\tan{A}}{1-\tan^2{A}}=\tan{2A}\)
\(=\int{2\theta.dx}\)
\(=2\int{\theta.dx}\)
\(=2\int{\tan^{-1}{x}dx}\) ➜Note \(\because \theta=\tan^{-1}{x}\)
\(=2\int{\tan^{-1}{x}.1dx}\)
\(=2\tan^{-1}{x}\int{1dx}-2\int{\left\{\frac{d}{dx}(\tan^{-1}{x})\int{1dx}\right\}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=2\tan^{-1}{x}.x-2\int{\left\{\frac{1}{1+x^2}.x\right\}dx}\) ➜Note \(\because \int{1dx}=x, \frac{d}{dx}(\tan^{-1}{x})=\frac{1}{1+x^2}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=2x\tan^{-1}{x}-2\int{\frac{1}{1+x^2}xdx}\)
\(=2x\tan^{-1}{x}-2\int{\frac{1}{t}\times{\frac{1}{2}}dt}\)
\(=2x\tan^{-1}{x}-\int{\frac{1}{t}dt}\)
\(=2x\tan^{-1}{x}-\ln{\left|t\right|}+c\) ➜Note \(\because \int{\frac{1}{x}dx}=\ln{|x|}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=2x\tan^{-1}{x}-\ln{|(1+x^2)|}+c\)

\(\int{\sin^{-1}{\sqrt{\frac{x}{a+x}}}dx}\)
\(=\int{\sin^{-1}{\sqrt{\frac{a\tan^2{\theta}}{a+a\tan^2{\theta}}}}.2a\tan{\theta}\sec^2{\theta}d\theta}\)
\(=\int{\sin^{-1}{\sqrt{\frac{a\tan^2{\theta}}{a(1+\tan^2{\theta})}}}.2a\tan{\theta}\sec^2{\theta}d\theta}\)
\(=\int{\sin^{-1}{\sqrt{\frac{\tan^2{\theta}}{1+\tan^2{\theta}}}}.2a\tan{\theta}\sec^2{\theta}d\theta}\)
\(=\int{\sin^{-1}{\sqrt{\frac{\tan^2{\theta}}{\sec^2{\theta}}}}.2a\tan{\theta}\sec^2{\theta}d\theta}\)
\(=\int{\sin^{-1}{\left(\frac{\tan{\theta}}{\sec{\theta}}\right)}.2a\tan{\theta}\sec^2{\theta}d\theta}\)
\(=\int{\sin^{-1}{\left(\frac{\frac{\sin{\theta}}{\cos{\theta}}}{\frac{1}{\cos{\theta}}}\right)}.2a\tan{\theta}\sec^2{\theta}d\theta}\)
\(=\int{\sin^{-1}{\left(\frac{\sin{\theta}}{\cos{\theta}}\times{\cos{\theta}}\right)}.2a\tan{\theta}\sec^2{\theta}d\theta}\)
\(=\int{\sin^{-1}{\left(\sin{\theta}\right)}.2a\tan{\theta}\sec^2{\theta}d\theta}\)
\(=\int{\theta.2a\tan{\theta}\sec^2{\theta}d\theta}\)
ধরি,
\(I=2a\int{\theta\tan{\theta}\sec^2{\theta}d\theta}\)
\(=2a\left[\theta\tan{\theta}\int{\sec^2{\theta}d\theta}-\int{\left\{\frac{d}{d\theta}(\theta\tan{\theta})\int{\sec^2{\theta}d\theta}\right\}d\theta}\right]\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=2a\left[\theta\tan{\theta}.\tan{\theta}-\int{(\tan{\theta}+\theta\sec^2{\theta})\tan{\theta}d\theta}\right]\)
\(=2a\left[\theta\tan^2{\theta}-\int{(\tan^2{\theta}+\theta\tan{\theta}\sec^2{\theta})d\theta}\right]\)
\(=2a\theta\tan^2{\theta}-2a\int{(\tan^2{\theta}+\theta\tan{\theta}\sec^2{\theta})d\theta}\)
\(=2a\theta\tan^2{\theta}-2a\int{\tan^2{\theta}d\theta}-2a\int{\theta\tan{\theta}\sec^2{\theta}d\theta}\)
\(I=2a\theta\tan^2{\theta}-2a\int{(\sec^2{\theta}-1)d\theta}-I\) ➜Note \(\because I=2a\int{\theta\tan{\theta}\sec^2{\theta}d\theta}\)
\(\Rightarrow I+I=2a\theta\tan^2{\theta}-2a\int{\sec^2{\theta}d\theta}+2a\int{1d\theta}\)
\(\Rightarrow 2I=2a\theta\tan^2{\theta}-2a\tan{\theta}+2a\theta+c_{1}\) ➜Note \(c_{1}\) যোগজীকরণ ধ্রুবক।
\(\Rightarrow 2I=2a\theta\tan^2{\theta}-2a\tan{\theta}-2a\theta+c_{1}\)
\(\Rightarrow I=a\theta\tan^2{\theta}-a\tan{\theta}+a\theta+\frac{1}{2}c_{1}\)
\(\Rightarrow 2a\int{\theta\tan{\theta}\sec^2{\theta}d\theta}=a\theta\tan^2{\theta}-a\tan{\theta}+a\theta+c\) ➜Note \(\because c=\frac{1}{2}c_{1}\)
\(\Rightarrow \int{\sin^{-1}{\sqrt{\frac{x}{a+x}}}dx}=\theta .a\tan^2{\theta}-a\tan{\theta}+a\theta+c\)
\(=\tan^{-1}{\left(\sqrt{\frac{x}{a}}\right)}.x-a\sqrt{\frac{x}{a}}+a\tan^{-1}{\left(\sqrt{\frac{x}{a}}\right)}+c\) ➜Note \(\because \theta=\tan^{-1}{\left(\sqrt{\frac{x}{a}}\right)}, a\tan^2{\theta}=x, \tan{\theta}=\sqrt{\frac{x}{a}}\)
\(=x\tan^{-1}{\left(\sqrt{\frac{x}{a}}\right)}-a\frac{\sqrt{x}}{\sqrt{a}}+a\tan^{-1}{\left(\sqrt{\frac{x}{a}}\right)}+c\)
\(=(x+a)\tan^{-1}{\left(\sqrt{\frac{x}{a}}\right)}-\sqrt{a}.\sqrt{a}\frac{\sqrt{x}}{\sqrt{a}}+c\)
\(=(x+a)\tan^{-1}{\left(\sqrt{\frac{x}{a}}\right)}-\sqrt{a}\sqrt{x}+c\)
\(=(x+a)\tan^{-1}{\left(\sqrt{\frac{x}{a}}\right)}-\sqrt{ax}+c\)

\(\int{\tan^{-1}{\sqrt{\frac{1-x}{1+x}}}dx}\)
\(=\int{\tan^{-1}{\sqrt{\frac{1-\cos{\theta}}{1+\cos{\theta}}}}dx}\)
\(=\int{\tan^{-1}{\sqrt{\frac{2\sin^2{\frac{\theta}{2}}}{2\cos^2{\frac{\theta}{2}}}}}dx}\) ➜Note \(\because 1-\cos{A}=2\sin^2{\frac{A}{2}}, 1+\cos{A}=2\cos^2{\frac{A}{2}}\)
\(=\int{\tan^{-1}{\sqrt{\frac{\sin^2{\frac{\theta}{2}}}{\cos^2{\frac{\theta}{2}}}}}dx}\)
\(=\int{\tan^{-1}{\frac{\sin{\frac{\theta}{2}}}{\cos{\frac{\theta}{2}}}}dx}\)
\(=\int{\tan^{-1}{\tan{\frac{\theta}{2}}}dx}\)
\(=\int{\frac{\theta}{2}dx}\)
\(=\int{\frac{1}{2}\theta.dx}\)
\(=\int{\frac{1}{2}\cos^{-1}{x}.dx}\) ➜Note \(\because \theta=\cos^{-1}{x}\)
\(=\frac{1}{2}\int{\cos^{-1}{x}.1dx}\)
\(=\frac{1}{2}\left[\cos^{-1}{x}\int{1dx}-\int{\left\{\frac{d}{dx}(\cos^{-1}{x})\int{1dx}\right\}dx}\right]\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{1}{2}\cos^{-1}{x}.x-\frac{1}{2}\int{\left(-\frac{1}{\sqrt{1-x^2}}\right).xdx}\) ➜Note \(\because \int{1dx}=x, \frac{d}{dx}(\cos^{-1}{x})=-\frac{1}{\sqrt{1-x^2}}\)
\(=\frac{1}{2}x\cos^{-1}{x}+\frac{1}{2}\int{\frac{x}{\sqrt{1-x^2}}dx}\)
\(=\frac{1}{2}x\cos^{-1}{x}+\frac{1}{2}\int{\frac{1}{\sqrt{1-x^2}}xdx}\)
\(=\frac{1}{2}x\cos^{-1}{x}+\frac{1}{2}\int{\frac{1}{t}\times{-tdt}}\)
\(=\frac{1}{2}x\cos^{-1}{x}-\frac{1}{2}\int{1dt}\)
\(=\frac{1}{2}x\cos^{-1}{x}-\frac{1}{2}t+c\) ➜Note \(\because \int{1dx}=x\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{2}x\cos^{-1}{x}-\frac{1}{2}\sqrt{1-x^2}+c\) ➜Note \(\because t=\sqrt{1-x^2}\)
\(=\frac{1}{2}(x\cos^{-1}{x}-\sqrt{1-x^2})+c\)

\(\int{\cos^{-1}{\sqrt{x}}dx}\)
\(=\int{\cos^{-1}{t}.2tdt}\)
\(=2\int{\cos^{-1}{t}.tdt}\)
\(=2\left[\cos^{-1}{t}\int{tdt}-\int{\left\{\frac{d}{dt}(\cos^{-1}{t})\int{tdt}\right\}dt}\right]\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=2\left[\cos^{-1}{t}.\frac{t^2}{2}-\int{\left(-\frac{1}{\sqrt{1-t^2}}\right).\frac{t^2}{2}dt}\right]\) ➜Note \(\because \int{xdx}=\frac{x^2}{2}, \frac{d}{dx}(\cos^{-1}{x})=-\frac{1}{\sqrt{1-x^2}}\)
\(=2\left[\frac{1}{2}t^2\cos^{-1}{t}+\frac{1}{2}\int{\frac{t^2}{\sqrt{1-t^2}}dt}\right]\)
\(=t^2\cos^{-1}{t}+\int{\frac{t^2}{\sqrt{1-t^2}}dt}\)
\(=t^2\cos^{-1}{t}+\int{\frac{1-(1-t^2)}{\sqrt{1-t^2}}dt}\)
\(=t^2\cos^{-1}{t}+\int{\left(\frac{1}{\sqrt{1-t^2}}-\frac{1-t^2}{\sqrt{1-t^2}}\right)dt}\)
\(=t^2\cos^{-1}{t}+\int{\frac{1}{\sqrt{1-t^2}}dt}-\int{\frac{1-t^2}{\sqrt{1-t^2}}dt}\)
\(=t^2\cos^{-1}{t}+\int{\frac{1}{\sqrt{1-t^2}}dt}-\int{\frac{\sqrt{1-t^2}.\sqrt{1-t^2}}{\sqrt{1-t^2}}dt}\)
\(=t^2\cos^{-1}{t}+\int{\frac{1}{\sqrt{1-t^2}}dt}-\int{\sqrt{1-t^2}dt}\)
\(=t^2\cos^{-1}{t}+\sin^{-1}{t}-\left\{\frac{t\sqrt{1-t^2}}{2}+\frac{1^2}{2}\sin^{-1}{\left(\frac{t}{1}\right)}\right\}+c\) ➜Note \(\because \int{\sqrt{a^2-x^2}dx}=\frac{x\sqrt{a^2-x^2}}{2}+\frac{a^2}{2}\sin^{-1}{\left(\frac{x}{a}\right)}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=t^2\cos^{-1}{t}+\sin^{-1}{t}-\left\{\frac{t\sqrt{1-t^2}}{2}+\frac{1}{2}\sin^{-1}{t}\right\}+c\)
\(=t^2\cos^{-1}{t}+\sin^{-1}{t}-\frac{t\sqrt{1-t^2}}{2}-\frac{1}{2}\sin^{-1}{t}+c\)
\(=x\cos^{-1}{\sqrt{x}}+\sin^{-1}{\sqrt{x}}-\frac{\sqrt{x}\sqrt{1-x}}{2}-\frac{1}{2}\sin^{-1}{\sqrt{x}}+c\) ➜Note \(\because t=\sqrt{x}\)
\(=x\cos^{-1}{\sqrt{x}}+\sin^{-1}{\sqrt{x}}-\frac{1}{2}\sin^{-1}{\sqrt{x}}-\frac{\sqrt{x}\sqrt{1-x}}{2}+c\)
\(=x\cos^{-1}{\sqrt{x}}+\frac{2-1}{2}\sin^{-1}{\sqrt{x}}-\frac{\sqrt{x}\sqrt{1-x}}{2}+c\)
\(=x\cos^{-1}{\sqrt{x}}+\frac{1}{2}\sin^{-1}{\sqrt{x}}-\frac{\sqrt{x(1-x)}}{2}+c\)

\(\int{\sin^{-1}{\sqrt{x}}dx}\)
\(=\int{\sin^{-1}{t}.2tdt}\)
\(=2\int{\sin^{-1}{t}.tdt}\)
\(=2\left[\sin^{-1}{t}\int{tdt}-\int{\left\{\frac{d}{dt}(\sin^{-1}{t})\int{tdt}\right\}dt}\right]\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=2\left[\sin^{-1}{t}.\frac{t^2}{2}-\int{\frac{1}{\sqrt{1-t^2}}.\frac{t^2}{2}dt}\right]\) ➜Note \(\because \int{xdx}=\frac{x^2}{2}, \frac{d}{dx}(\sin^{-1}{x})=\frac{1}{\sqrt{1-x^2}}\)
\(=2\left[\frac{1}{2}t^2\sin^{-1}{t}-\frac{1}{2}\int{\frac{t^2}{\sqrt{1-t^2}}dt}\right]\)
\(=t^2\sin^{-1}{t}-\int{\frac{t^2}{\sqrt{1-t^2}}dt}\)
\(=t^2\sin^{-1}{t}-\int{\frac{1-(1-t^2)}{\sqrt{1-t^2}}dt}\)
\(=t^2\sin^{-1}{t}-\int{\left(\frac{1}{\sqrt{1-t^2}}-\frac{1-t^2}{\sqrt{1-t^2}}\right)dt}\)
\(=t^2\sin^{-1}{t}-\int{\frac{1}{\sqrt{1-t^2}}dt}+\int{\frac{1-t^2}{\sqrt{1-t^2}}dt}\)
\(=t^2\sin^{-1}{t}-\int{\frac{1}{\sqrt{1-t^2}}dt}+\int{\frac{\sqrt{1-t^2}.\sqrt{1-t^2}}{\sqrt{1-t^2}}dt}\)
\(=t^2\sin^{-1}{t}-\int{\frac{1}{\sqrt{1-t^2}}dt}+\int{\sqrt{1-t^2}dt}\)
\(=t^2\sin^{-1}{t}-\sin^{-1}{t}+\left\{\frac{t\sqrt{1-t^2}}{2}+\frac{1^2}{2}\sin^{-1}{\left(\frac{t}{1}\right)}\right\}+c\) ➜Note \(\because \int{\sqrt{a^2-x^2}dx}=\frac{x\sqrt{a^2-x^2}}{2}+\frac{a^2}{2}\sin^{-1}{\left(\frac{x}{a}\right)}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=t^2\sin^{-1}{t}-\sin^{-1}{t}+\left\{\frac{t\sqrt{1-t^2}}{2}+\frac{1}{2}\sin^{-1}{t}\right\}+c\)
\(=t^2\sin^{-1}{t}-\sin^{-1}{t}+\frac{t\sqrt{1-t^2}}{2}+\frac{1}{2}\sin^{-1}{t}+c\)
\(=x\sin^{-1}{\sqrt{x}}-\sin^{-1}{\sqrt{x}}+\frac{\sqrt{x}\sqrt{1-x}}{2}+\frac{1}{2}\sin^{-1}{\sqrt{x}}+c\) ➜Note \(\because t=\sqrt{x}\)
\(=x\sin^{-1}{\sqrt{x}}-\sin^{-1}{\sqrt{x}}+\frac{1}{2}\sin^{-1}{\sqrt{x}}+\frac{\sqrt{x}\sqrt{1-x}}{2}+c\)
\(=x\sin^{-1}{\sqrt{x}}-\frac{2-1}{2}\sin^{-1}{\sqrt{x}}+\frac{\sqrt{x}\sqrt{1-x}}{2}+c\)
\(=x\sin^{-1}{\sqrt{x}}-\frac{1}{2}\sin^{-1}{\sqrt{x}}+\frac{\sqrt{x(1-x)}}{2}+c\)
\(=\left(x-\frac{1}{2}\right)\sin^{-1}{\sqrt{x}}+\frac{\sqrt{x(1-x)}}{2}+c\)

\(\int{\tan^{-1}{\sqrt{x}}dx}\)
\(=\int{\tan^{-1}{t}.2tdt}\)
\(=2\int{\tan^{-1}{t}.tdt}\)
\(=2\left[\tan^{-1}{t}\int{tdt}-\int{\left\{\frac{d}{dt}(\tan^{-1}{t})\int{tdt}\right\}dt}\right]\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=2\left[\tan^{-1}{t}.\frac{t^2}{2}-\int{\frac{1}{1+t^2}.\frac{t^2}{2}dt}\right]\) ➜Note \(\because \int{xdx}=\frac{x^2}{2}, \frac{d}{dx}(\tan^{-1}{x})=\frac{1}{1+x^2}\)
\(=2\left[\frac{1}{2}t^2\tan^{-1}{t}-\frac{1}{2}\int{\frac{t^2}{1+t^2}dt}\right]\)
\(=t^2\tan^{-1}{t}-\int{\frac{t^2}{1+t^2}dt}\)
\(=t^2\tan^{-1}{t}-\int{\frac{1+t^2-1}{1+t^2}dt}\)
\(=t^2\tan^{-1}{t}-\int{\left(\frac{1+t^2}{1+t^2}-\frac{1}{1+t^2}\right)dt}\)
\(=t^2\tan^{-1}{t}-\int{1dt}+\int{\frac{1}{1+t^2}dt}\)
\(=t^2\tan^{-1}{t}-t+\tan^{-1}{t}+c\) ➜Note \(\because \int{1dx}=x, \int{\frac{1}{1+x^2}dx}=\tan^{-1}{x}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=x\tan^{-1}{\sqrt{x}}-\sqrt{x}+\tan^{-1}{\sqrt{x}}+c\) ➜Note \(\because t=\sqrt{x}\)
\(=x\tan^{-1}{\sqrt{x}}+\tan^{-1}{\sqrt{x}}-\sqrt{x}+c\)
\(=(x+1)\tan^{-1}{\sqrt{x}}-\sqrt{x}+c\)

\(\int{\sec^{-1}{x}dx}\)
\(=\int{\sec^{-1}{x}.1dx}\)
\(=\sec^{-1}{x}\int{1dx}-\int{\left\{\frac{d}{dx}(\sec^{-1}{x})\int{1dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\sec^{-1}{x}.x-\int{\frac{1}{x\sqrt{x^2-1}}.xdx}\) ➜Note \(\because \int{1dx}=x, \frac{d}{dx}(\sec^{-1}{x})=\frac{1}{x\sqrt{x^2-1}}\)
\(=x\sec^{-1}{x}-\int{\frac{1}{\sqrt{x^2-1}}dx}\)
\(=x\sec^{-1}{x}-\ln{|\sqrt{x^2-1}+x|}+c\) ➜Note \(\because \int{\frac{1}{\sqrt{x^2-a^2}}dx}=\ln{|\sqrt{x^2-a^2}+x|}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=x\sec^{-1}{x}-\ln{|x+\sqrt{x^2-1}|}+c\)

\(\int{x^3e^{x^2}dx}\)
\(=\int{x^2e^{x^2}.xdx}\)
\(=\int{te^{t}.\frac{1}{2}dt}\)
\(=\frac{1}{2}\int{te^{t}dt}\)
\(=\frac{1}{2}\left[t\int{e^{t}dt}-\int{\left\{\frac{d}{dt}(t)\int{e^tdt}\right\}dt}\right]\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{1}{2}\left(te^{t}-\int{1.e^tdt}\right)\) ➜Note \(\because \int{e^{x}dx}=e^x, \frac{d}{dx}(x)=1\)
\(=\frac{1}{2}\left(te^{t}-\int{e^tdt}\right)\)
\(=\frac{1}{2}\left(te^{t}-e^t\right)+c\) ➜Note\(\because \int{e^{x}dx}=e^x\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{2}e^t(t-1)+c\)
\(=\frac{1}{2}e^{x^2}(x^2-1)+c\) ➜Note\(\because t=x^2\)

\(\int{e^{2t}\cos{\left(e^{t}\right)}dt}\)
\(=\int{e^{t}.e^{t}\cos{\left(e^{t}\right)}dt}\)
\(=\int{e^{t}\cos{\left(e^{t}\right)}e^{t}dt}\)
\(=\int{x\cos{x}dx}\)
\(=x\int{\cos{x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\cos{x}dx}\right\}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=x\sin{x}-\int{1.\sin{x}dx}\) ➜Note \(\because \int{\cos{x}dx}=\sin{x}, \frac{d}{dx}(x)=1\)
\(=x\sin{x}-\int{\sin{x}dx}\)
\(=x\sin{x}+\cos{x}+c\) ➜Note\(\because \int{\sin{x}dx}=-\cos{x}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=e^{t}\sin{\left(e^{t}\right)}+\cos{\left(e^{t}\right)}+c\) ➜Note\(\because x=e^{t}\)

\(\int{y\sin^{-1}{(y^2)}dx}\)
\(=\int{\sin^{-1}{(y^2)}.ydy}\)
\(=\int{\sin^{-1}{t}.\frac{1}{2}dt}\)
\(=\frac{1}{2}\int{\sin^{-1}{t}.1dt}\)
\(=\frac{1}{2}\left[\sin^{-1}{t}\int{1dt}-\int{\left\{\frac{d}{dt}(\sin^{-1}{t})\int{1dt}\right\}dt}\right]\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{1}{2}\left(\sin^{-1}{t}.t-\int{\frac{1}{\sqrt{1-t^2}}.tdt}\right)\) ➜Note \(\because \int{1dx}=x, \frac{d}{dx}(\sin^{-1}{x})=\frac{1}{\sqrt{1-x^2}}\)
\(=\frac{1}{2}\left(t\sin^{-1}{t}-\int{\frac{t}{\sqrt{1-t^2}}dt}\right)\)
\(=\frac{1}{2}\left(t\sin^{-1}{t}+\frac{1}{2}\int{\frac{-2t}{\sqrt{1-t^2}}dt}\right)\)
\(=\frac{1}{2}\left(t\sin^{-1}{t}+\frac{1}{2}\int{\frac{1}{\sqrt{1-t^2}}\times{-2tdt}}\right)\)
\(=\frac{1}{2}\left(t\sin^{-1}{t}+\frac{1}{2}\int{\frac{1}{\sqrt{z}}\times{dz}}\right)\)
\(=\frac{1}{2}\left(t\sin^{-1}{t}+\frac{1}{2}\int{z^{-\frac{1}{2}}dz}\right)\)
\(=\frac{1}{2}\left(t\sin^{-1}{t}+\frac{1}{2}.\frac{z^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}\right)+c\) ➜Note \(\because \int{x^ndx}=\frac{x^{n+1}}{n+1}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{2}\left(t\sin^{-1}{t}+\frac{1}{2}.\frac{z^{\frac{-1+2}{2}}}{\frac{-1+2}{2}}\right)+c\)
\(=\frac{1}{2}\left(t\sin^{-1}{t}+\frac{1}{2}.\frac{z^{\frac{1}{2}}}{\frac{1}{2}}\right)+c\)
\(=\frac{1}{2}\left(t\sin^{-1}{t}+z^{\frac{1}{2}}\right)+c\)
\(=\frac{1}{2}\left(t\sin^{-1}{t}+\sqrt{z}\right)+c\)
\(=\frac{1}{2}\left(t\sin^{-1}{t}+\sqrt{1-t^2}\right)+c\) ➜Note\(\because z=1-t^2\)
\(=\frac{1}{2}\left\{y^2\sin^{-1}{(y^2)}+\sqrt{1-y^4}\right\}+c\) ➜Note\(\because t=y^2\)

\(\int{\frac{\ln{|\sec^{-1}{x}|}}{x\sqrt{x^2-1}}dx}\)
\(=\int{\ln{|\sec^{-1}{x}|}.\frac{1}{x\sqrt{x^2-1}}dx}\)
\(=\int{\ln{|t|}dt}\)
\(=\int{\ln{|t|}.1dt}\)
\(=\ln{|t|}\int{1dt}-\int{\left\{\frac{d}{dt}(\ln{|t|})\int{1dt}\right\}dt}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\ln{|t|}.t-\int{\frac{1}{t}.tdt}\) ➜Note \(\because \int{1dx}=x, \frac{d}{dx}(\ln{|x|})=\frac{1}{t}\)
\(=t\ln{|t|}-\int{1dt}\)
\(=t\ln{|t|}-t+c\) ➜Note \(\because \int{1dx}=x\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\sec^{-1}{x}\ln{|\sec^{-1}{x}|}-\sec^{-1}{x}+c\) ➜Note \(\because t=\sec^{-1}{x}\)
\(=\sec^{-1}{x}(\ln{|\sec^{-1}{x}|}-1)+c\)

\(\int{e^{x}(\sin{x}+\cos{x})dx}\)
\(=\int{e^{x}\sin{x}dx}+\int{e^{x}\cos{x})dx}\)
\(=\sin{x}\int{e^{x}dx}-\int{\left\{\frac{d}{dx}(\sin{x})\int{e^xdx}\right\}dx}+\int{e^{x}\cos{x})dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\sin{x}.e^{x}-\int{\cos{x}.e^xdx}+\int{e^{x}\cos{x})dx}\) ➜Note \(\because \int{e^xdx}=e^x, \frac{d}{dx}(\sin{x})=\cos{x}\)
\(=e^{x}\sin{x}-\int{e^{x}\cos{x}dx}+\int{e^{x}\cos{x})dx}\)
\(=e^{x}\sin{x}+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।

\(\int{\frac{e^{x}}{x}(1+x\ln{|x|})dx}\)
\(=\int{\frac{e^{x}}{x}dx}+\int{e^{x}\ln{|x|})dx}\)
\(=\int{\frac{e^{x}}{x}dx}+\int{\ln{|x|})e^{x}dx}\)
\(=\int{\frac{e^{x}}{x}dx}+\ln{|x|}\int{e^{x}dx}-\int{\left\{\frac{d}{dx}(\ln{|x|})\int{e^xdx}\right\}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\int{\frac{e^{x}}{x}dx}+\ln{|x|}.e^{x}-\int{\frac{1}{x}e^xdx}\) ➜Note \(\because \int{e^{x}dx}=e^x, \frac{d}{dx}(\ln{|x|})=\frac{1}{x}\)
\(=\int{\frac{e^{x}}{x}dx}+e^{x}\ln{|x|}-\int{\frac{e^{x}}{x}dx}\)
\(=e^{x}\ln{|x|}+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।

\(\int{e^{x}\left\{\frac{1}{1-x}+\frac{1}{(1-x)^2}\right\}dx}\)
\(=\int{e^{x}\frac{1}{1-x}dx}+\int{e^{x}\frac{1}{(1-x)^2}dx}\)
\(=\frac{1}{1-x}\int{e^{x}dx}-\int{\left\{\frac{d}{dx}(\frac{1}{1-x})\int{e^xdx}\right\}dx}+\int{e^{x}\frac{1}{(1-x)^2}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{1}{1-x}.e^{x}-\int{\left(-\frac{1}{(1-x)^2}\right)\times{\frac{d}{dx}(1-x)}.e^xdx}+\int{e^{x}\frac{1}{(1-x)^2}dx}\) ➜Note \(\because \int{e^{x}dx}=e^x, \frac{d}{dx}(\frac{1}{x})=-\frac{1}{x^2}\)
\(=\frac{e^{x}}{1-x}-\int{\left(-\frac{1}{(1-x)^2}\right)\times{-1}.e^xdx}+\int{e^{x}\frac{1}{(1-x)^2}dx}\)
\(=\frac{e^{x}}{1-x}-\int{e^{x}\frac{1}{(1-x)^2}dx}+\int{e^{x}\frac{1}{(1-x)^2}dx}\)
\(=\frac{e^{x}}{1-x}+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।

\(\int{e^{5x}\left\{5\ln{|x|}+\frac{1}{x}\right\}dx}\)
\(=\int{5e^{5x}\ln{|x|}dx}+\int{e^{5x}\frac{1}{x}dx}\)
\(=5\int{e^{5x}\ln{|x|}dx}+\int{\frac{1}{x}e^{5x}dx}\)
\(=5\ln{|x|}\int{e^{5x}dx}-5\int{\left\{\frac{d}{dx}(\ln{|x|})\int{e^{5x}dx}\right\}dx}+\int{\frac{1}{x}e^{5x}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=5\ln{|x|}\frac{e^{5x}}{5}-5\int{\frac{1}{x}.\frac{e^{5x}}{5}dx}+\int{\frac{1}{x}e^{5x}dx}\) ➜Note \(\because \int{e^{ax}dx}=\frac{e^{ax}}{a}, \frac{d}{dx}(\ln{|x|})=\frac{1}{x}\)
\(=\ln{|x|}e^{5x}-\int{\frac{1}{x}e^{5x}dx}+\int{\frac{1}{x}e^{5x}dx}\)
\(=e^{5x}\ln{|x|}+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।

\(\int{e^{-x}\left\{\frac{1}{x}+\frac{1}{x^2}\right\}dx}\)
\(=\int{e^{-x}\frac{1}{x}dx}+\int{e^{-x}\frac{1}{x^2}dx}\)
\(=\frac{1}{x}\int{e^{-x}dx}-\int{\left\{\frac{d}{dx}\left(\frac{1}{x}\right)\int{e^{-x}dx}\right\}dx}+\int{e^{-x}\frac{1}{x^2}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{1}{x}\times{\frac{e^{-x}}{-1}}-\int{\left(-\frac{1}{x^2}\right)\times{\frac{e^{-x}}{-1}}dx}+\int{e^{-x}\frac{1}{x^2}dx}\) ➜Note \(\because \int{e^{ax}dx}=\frac{e^{ax}}{a}, \frac{d}{dx}(\frac{1}{x})=-\frac{1}{x^2}\)
\(=-\frac{1}{x}e^{-x}-\int{\frac{1}{x^2}e^{-x}dx}+\int{e^{-x}\frac{1}{x^2}dx}\)
\(=-\frac{1}{x}e^{-x}-\int{e^{-x}\frac{1}{x^2}dx}+\int{e^{-x}\frac{1}{x^2}dx}\)
\(=-\frac{1}{x}e^{-x}+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।
\(=-\frac{e^{-x}}{x}+c\)

\(\int{\frac{xe^{x}}{(x+1)^2}dx}\)
\(=\int{\frac{e^{x}x}{(x+1)^2}dx}\)
\(=\int{\frac{e^{x}(x+1-1)}{(x+1)^2}dx}\)
\(=\int{\frac{e^{x}(x+1)-e^{x}}{(x+1)^2}dx}\)
\(=\int{\left\{\frac{e^{x}(x+1)}{(x+1)^2}-\frac{e^{x}}{(x+1)^2}\right\}dx}\)
\(=\int{\left\{\frac{e^{x}}{x+1}-\frac{e^{x}}{(x+1)^2}\right\}dx}\)
\(=\int{\frac{e^{x}}{x+1}dx}-\int{\frac{e^{x}}{(x+1)^2}dx}\)
\(=\int{\frac{1}{x+1}e^{x}dx}-\int{\frac{e^{x}}{(x+1)^2}dx}\)
\(=\frac{1}{x+1}\int{e^{x}dx}-\int{\left\{\frac{d}{dx}\left(\frac{1}{x+1}\right)\int{e^{x}dx}\right\}dx}-\int{\frac{e^{x}}{(x+1)^2}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{1}{x+1}e^x-\int{\left(-\frac{1}{(x+1)^2}\right)e^{x}dx}-\int{\frac{e^{x}}{(x+1)^2}dx}\) ➜Note \(\because \int{e^{x}dx}=e^{x}, \frac{d}{dx}(\frac{1}{x+1})=-\frac{1}{(x+1)^2}\)
\(=\frac{e^x}{x+1}+\int{\frac{e^{x}}{(x+1)^2}dx}-\int{\frac{e^{x}}{(x+1)^2}dx}\)
\(=\frac{e^x}{x+1}+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।

\(\int{e^{x}(\tan{x}-\ln{|\cos{x}|})dx}\)
\(=\int{e^{x}\tan{x}dx}-\int{e^{x}\ln{|\cos{x}|}dx}\)
\(=\int{e^{x}\tan{x}dx}-\left[\ln{|\cos{x}|}\int{e^{x}dx}-\int{\left\{\frac{d}{dx}(\ln{|\cos{x}|})\int{e^xdx}\right\}dx}\right]\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\int{e^{x}\tan{x}dx}-\left[\ln{|\cos{x}|}.e^{x}-\int{\frac{1}{\cos{x}}.\frac{d}{dx}(\cos{x}).e^xdx}\right]\) ➜Note \(\because \int{e^{x}dx}=e^{x}, \frac{d}{dx}(\ln{|x|})=\frac{1}{x}\)
\(=\int{e^{x}\tan{x}dx}-\left[e^{x}\ln{|\cos{x}|}-\int{e^x\frac{1}{\cos{x}}.(-\sin{x})dx}\right]\) ➜Note \(\because \frac{d}{dx}(\cos{x})=-\sin{x}\)
\(=\int{e^{x}\tan{x}dx}-\left[e^{x}\ln{|\cos{x}|}+\int{e^x\frac{\sin{x}}{\cos{x}}dx}\right]\)
\(=\int{e^{x}\tan{x}dx}-e^{x}\ln{|\cos{x}|}-\int{e^x\tan{x}dx}\)
\(=-e^{x}\ln{|\cos{x}|}+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।

\(\int{cosec^3{x}dx}\)
ধরি,
\(I=\int{cosec^3{x}dx}\)
\(=\int{cosec \ {x}. \ cosec^2{x}dx}\)
\(=cosec \ {x}\int{cosec^2{x}dx}-\int{\left\{\frac{d}{dx}(cosec \ {x})\int{cosec^2{x}dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=cosec \ {x}\times{-\cot{x}}-\int{-cosec \ {x}\cot{x}\times{-\cot{x}}dx}\) ➜Note \(\because \int{cosec^2{x}dx}=-\cot{x}, \frac{d}{dx}(cosec \ {x})=-cosec \ {x}\cot{x}\)
\(=-cosec \ {x}\cot{x}-\int{cosec \ {x}\cot^2{x}dx}\)
\(=-cosec \ {x}\cot{x}-\int{cosec \ {x}(cosec^2{x}-1)dx}\) ➜Note \(\because \cot^2{x}=cosec^2{x}-1\)
\(=-cosec \ {x}\cot{x}-\int{(cosec^3{x}-cosec \ {x})dx}\)
\(=-cosec \ {x}\cot{x}-\int{cosec^3{x}dx}+\int{cosec \ {x}dx}\)
\(=-cosec \ {x}\cot{x}-\int{cosec^3{x}dx}+\ln{\left|\tan{\left(\frac{x}{2}\right)}\right|}+c_{1}\) ➜Note \(\because \int{cosec \ {x}dx}=\ln{\left|\tan{\left(\frac{x}{2}\right)}\right|}\), \(c_{1}\) যোগজীকরণ ধ্রুবক।
\(\Rightarrow I=-cosec \ {x}\cot{x}-I+\ln{\left|\tan{\left(\frac{x}{2}\right)}\right|}+c_{1}\) ➜Note \(\because \int{cosec^3{x}dx}=I\)
\(\Rightarrow I+I=-cosec \ {x}\cot{x}+\ln{\left|\tan{\left(\frac{x}{2}\right)}\right|}+c_{1}\)
\(\Rightarrow 2I=-cosec \ {x}\cot{x}+\ln{\left|\tan{\left(\frac{x}{2}\right)}\right|}+c_{1}\)
\(\Rightarrow I=-\frac{1}{2} \ cosec \ {x}\cot{x}+\frac{1}{2}\ln{\left|\tan{\left(\frac{x}{2}\right)}\right|}+\frac{1}{2}c_{1}\)
\(\therefore \int{cosec^3{x}dx}=-\frac{1}{2} \ cosec \ {x}\cot{x}+\frac{1}{2}\ln{\left|\tan{\left(\frac{x}{2}\right)}\right|}+c\) ➜Note \(c=\frac{1}{2}c_{1}\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{2}\left\{\ln{\left|\tan{\left(\frac{x}{2}\right)}\right|}-cosec \ {x}\cot{x}\right\}+c\)

\(\int{\sin{(\ln{|x|})}dx}\)
ধরি,
\(I=\int{\sin{(\ln{|x|})}dx}\)
\(=\int{\sin{(\ln{|x|})}.1dx}\)
\(=\sin{(\ln{|x|})}\int{1dx}-\int{\left\{\frac{d}{dx}(\sin{(\ln{|x|})})\int{1dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\sin{(\ln{|x|})}.x-\int{\cos{(\ln{|x|}}\frac{d}{dx}(\ln{|x|}).xdx}\) ➜Note \(\because \int{1dx}=x, \frac{d}{dx}(\sin{x})=\cos{x}\)
\(=x\sin{(\ln{|x|})}-\int{\cos{(\ln{|x|}}\frac{1}{x}.xdx}\) ➜Note \(\because \frac{d}{dx}(\ln{|x|})=\frac{1}{x}\)
\(=x\sin{(\ln{|x|})}-\int{\cos{(\ln{|x|}}.1dx}\)
\(=x\sin{(\ln{|x|})}-\left[\cos{(\ln{|x|})}\int{1dx}-\int{\left\{\frac{d}{dx}(\cos{(\ln{|x|})})\int{1dx}\right\}dx}\right]\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=x\sin{(\ln{|x|})}-\left[\cos{(\ln{|x|})}.x-\int{(-\sin{(\ln{|x|})})\frac{d}{dx}(\ln{|x|}).xdx}\right]\) ➜Note \(\because \int{1dx}=x, \frac{d}{dx}(\cos{x})=-\sin{x}\)
\(=x\sin{(\ln{|x|})}-\left[x\cos{(\ln{|x|})}+\int{\sin{(\ln{|x|})}\frac{1}{x}.xdx}\right]\) ➜Note \(\because \frac{d}{dx}(\ln{|x|})=\frac{1}{x}\)
\(=x\sin{(\ln{|x|})}-\left[x\cos{(\ln{|x|})}+\int{\sin{(\ln{|x|})}dx}\right]\)
\(=x\sin{(\ln{|x|})}-x\cos{(\ln{|x|})}-\int{\sin{(\ln{|x|})}dx}\)
\(I=x\sin{(\ln{|x|})}-x\cos{(\ln{|x|})}-I\) ➜Note \(\because \int{\sin{(\ln{|x|})}dx}=I\)
\(\Rightarrow I+I=x\sin{(\ln{|x|})}-x\cos{(\ln{|x|})}\)
\(\Rightarrow 2I=x\sin{(\ln{|x|})}-x\cos{(\ln{|x|})}\)
\(\Rightarrow I=\frac{1}{2}\left\{x\sin{(\ln{|x|})}-x\cos{(\ln{|x|})}\right\}+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।
\(\therefore \int{\sin{(\ln{|x|})}dx}=\frac{1}{2}x\left\{\sin{(\ln{|x|})}-\cos{(\ln{|x|})}\right\}+c\)

\(\int{\frac{e^{x}(x^2+1)}{(x+1)^2}dx}\)
\(=\int{\frac{e^{x}\{x^2+2x+1-2(x+1)+2\}}{(x+1)^2}dx}\)
\(=\int{\frac{e^{x}\{(x+1)^2-2(x+1)+2\}}{(x+1)^2}dx}\)
\(=\int{\frac{e^{x}(x+1)^2-2e^{x}(x+1)+2e^{x}}{(x+1)^2}dx}\)
\(=\int{\frac{e^{x}(x+1)^2}{(x+1)^2}dx}-2\int{\frac{e^{x}(x+1)}{(x+1)^2}dx}+2\int{\frac{e^{x}}{(x+1)^2}dx}\)
\(=\int{e^{x}dx}-2\int{\frac{e^{x}}{x+1}dx}+2\int{\frac{e^{x}}{(x+1)^2}dx}\)
\(=e^{x}-2\int{\frac{1}{x+1}.e^{x}dx}+2\int{\frac{e^{x}}{(x+1)^2}dx}\)
\(=e^{x}-2\left[\frac{1}{x+1}\int{e^{x}dx}-\int{\left\{\frac{d}{dx}\left(\frac{1}{x+1}\right)\int{e^xdx}\right\}dx}\right]+2\int{\frac{e^{x}}{(x+1)^2}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=e^{x}-2\left[\frac{1}{x+1}e^{x}-\int{\left\{-\frac{1}{(x+1)^2}\right\}e^xdx}\right]+2\int{\frac{e^{x}}{(x+1)^2}dx}\) ➜Note \(\because \int{e^xdx}=e^x, \frac{d}{dx}\left(\frac{1}{x}\right)=-\frac{1}{x^2}\)
\(=e^{x}-2\left[\frac{1}{x+1}e^{x}+\int{\frac{1}{(x+1)^2}e^xdx}\right]+2\int{\frac{e^{x}}{(x+1)^2}dx}\)
\(=e^{x}-\frac{2}{x+1}e^{x}-2\int{\frac{e^x}{(x+1)^2}dx}+2\int{\frac{e^{x}}{(x+1)^2}dx}\)
\(=e^{x}-\frac{2}{x+1}e^{x}+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।
\(=e^{x}\left(1-\frac{2}{x+1}\right)+c\)
\(=e^{x}\frac{x+1-2}{x+1}+c\)
\(=e^{x}\frac{x-1}{x+1}+c\)
\(=\frac{e^{x}(x-1)}{x+1}+c\)

\(\int{x\sec^2{3x}dx}\)
\(=x\int{\sec^2{3x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\sec^2{3x}dx}\right\}dx}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=x.\frac{\tan{3x}}{3}-\int{1.\frac{\tan{3x}}{3}dx}\) ➜Note \(\because \int{\sec^2{ax}dx}=\frac{\tan{ax}}{a}, \frac{d}{dx}(x)=1\)
\(=\frac{1}{3}x\tan{3x}-\frac{1}{3}\int{\tan{3x}dx}\)
\(=\frac{1}{3}x\tan{3x}-\frac{1}{3}\frac{\ln{|\sec{3x}|}}{3}+c\) ➜Note \(\because \int{\tan{ax}dx}=\frac{\ln{|\sec{ax}|}}{a}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{1}{3}x\tan{3x}-\frac{1}{9}\ln{|\sec{3x}|}+c\)
\(=\frac{1}{9}(3x\tan{3x}-\ln{|\sec{3x}|})+c\)

\(\int{x\tan^2{x}dx}\)
\(=\int{x(\sec^2{x}-1)dx}\)
\(=\int{(x\sec^2{x}-x)dx}\)
\(=\int{x\sec^2{x}dx}-\int{xdx}\)
\(=x\int{\sec^2{x}dx}-\int{\left\{\frac{d}{dx}(x)\int{\sec^2{x}dx}\right\}dx}-\frac{x^2}{2}\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}, \int{xdx}=\frac{x^2}{2}\)
\(=x.\tan{x}-\int{1.\tan{x}dx}-\frac{1}{2}x^2\) ➜Note \(\because \int{\sec^2{x}dx}=\tan{x}, \frac{d}{dx}(x)=1\)
\(=x\tan{x}-\int{\tan{x}dx}-\frac{1}{2}x^2\)
\(=x\tan{x}-\ln{|\sec{x}|}-\frac{1}{2}x^2+c\) ➜Note \(\because \int{\tan{x}dx}=\ln{|\sec{x}|}\) এবং \(c\) যোগজীকরণ ধ্রুবক।

\(\int{e^{-3x}\cos{4x}dx}\)
\(=\frac{e^{-3x}}{(-3)^2+4^2}(-3\cos{4x}+4\sin{4x})+c\) ➜Note \(\because \int{e^{ax}\cos{bx}dx}=\frac{e^{ax}}{a^2+b^2}(a\cos{bx}+b\sin{bx})\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{e^{-3x}}{9+16}(4\sin{4x}-3\cos{4x})+c\)
\(=\frac{1}{25}e^{-3x}(4\sin{4x}-3\cos{4x})+c\)

\(\int{\log_{10}{x}dx}\)
\(=\int{\log_{10}{e}\times{\ln{|x|}}dx}\) ➜Note \(\because \log_{a}{x}=\log_{a}{e}\times{\ln{|x|}}\)
\(=\log_{10}{e}\int{\ln{|x|}.1dx}\)
\(=\log_{10}{e}\left[\ln{|x|}\int{1dx}-\int{\left\{\frac{d}{dx}(\ln{|x|})\int{1dx}\right\}dx}\right]\) ➜Note \(\because \int{uvdx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\log_{10}{e}\left[\ln{|x|}.x-\int{\frac{1}{x}.xdx}\right]\) ➜Note \(\because \int{1dx}=x, \frac{d}{dx}(\ln{|x|})=\frac{1}{x}\)
\(=\log_{10}{e}\left[x\ln{|x|}-\int{1dx}\right]\)
\(=\log_{10}{e}\left[x\ln{|x|}-x\right]+c\) ➜Note \(\because \int{1dx}=x\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=x\log_{10}{e}\times{\ln{|x|}}-x\log_{10}{e}+c\)
\(=x\log_{10}{x}-x\log_{10}{e}+c\) ➜Note \(\because \log_{a}{e}\times{\ln{|x|}}=\log_{a}{x}\)

\(\int{e^{-2x}\left(\frac{1}{x}-2\ln{|x|}\right)dx}\)
\(=\int{\left(e^{-2x}\frac{1}{x}-2e^{-2x}\ln{|x|}\right)dx}\)
\(=\int{e^{-2x}\frac{1}{x}dx}-2\int{\ln{|x|}.e^{-2x}dx}\)
\(=\int{e^{-2x}\frac{1}{x}dx}-2\left[\ln{|x|}\int{e^{-2x}dx}-\int{\left\{\frac{d}{dx}(\ln{|x|})\int{e^{-2x}dx}\right\}dx}\right]\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\int{e^{-2x}\frac{1}{x}dx}-2\left[\ln{|x|}.\frac{e^{-2x}}{-2}-\int{\frac{1}{x}.\frac{e^{-2x}}{-2}dx}\right]\) ➜Note \(\because \int{e^{ax}dx}=\frac{e^{ax}}{a}, \frac{d}{dx}(\ln{|x|})=\frac{1}{x}\)
\(=\int{e^{-2x}\frac{1}{x}dx}-2\left[-\frac{1}{2}e^{-2x}\ln{|x|}+\frac{1}{2}\int{e^{-2x}\frac{1}{x}dx}\right]\)
\(=\int{e^{-2x}\frac{1}{x}dx}+e^{-2x}\ln{|x|}-\int{e^{-2x}\frac{1}{x}dx}\)
\(=e^{-2x}\ln{|x|}+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।

\(\int{\{\sin{(\ln{|x|})}+\cos{(\ln{|x|})}\}dx}\)
\(=\int{\sin{(\ln{|x|})}dx}+\int{\cos{(\ln{|x|})}dx}\)
\(=\int{\sin{(\ln{|x|})}.1dx}+\int{\cos{(\ln{|x|})}dx}\)
\(=\sin{(\ln{|x|})}\int{1dx}-\int{\left[\frac{d}{dx}\{\sin{(\ln{|x|})}\}\int{1dx}\right]dx}+\int{\cos{(\ln{|x|})}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\sin{(\ln{|x|})}.x-\int{\cos{(\ln{|x|})}.\frac{d}{dx}(\ln{|x|}).xdx}+\int{\cos{(\ln{|x|})}dx}\) ➜Note \(\because \frac{d}{dx}(\sin{x})=\cos{x}, \int{1dx}=x\)
\(=x\sin{(\ln{|x|})}-\int{\cos{(\ln{|x|})}.\frac{1}{x}.xdx}+\int{\cos{(\ln{|x|})}dx}\) ➜Note \(\because \frac{d}{dx}(\ln{|x|})=\frac{1}{x}\)
\(=x\sin{(\ln{|x|})}-\int{\cos{(\ln{|x|})}dx}+\int{\cos{(\ln{|x|})}dx}\)
\(=x\sin{(\ln{|x|})}+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।

\(\int{e^x\{\ln{|\sec{x}+\tan{x}|}+\sec{x}\}dx}\)
\(=\int{\{e^x\ln{|\sec{x}+\tan{x}|}+e^x\sec{x}\}dx}\)
\(=\int{e^x\ln{|\sec{x}+\tan{x}|}dx}+\int{e^x\sec{x}dx}\)
\(=\ln{|\sec{x}+\tan{x}|}\int{e^xdx}-\int{\left\{\frac{d}{dx}(\ln{|\sec{x}+\tan{x}|})\int{e^xdx}\right\}dx}+\int{e^x\sec{x}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\ln{|\sec{x}+\tan{x}|}.e^x-\int{\frac{1}{\sec{x}+\tan{x}}.\frac{d}{dx}(\sec{x}+\tan{x}).e^xdx}+\int{e^x\sec{x}dx}\) ➜Note \(\because \int{e^xdx}=e^{x}, \frac{d}{dx}(\ln{|x|})=\frac{1}{x}\)
\(=e^x\ln{|\sec{x}+\tan{x}|}-\int{\frac{1}{\sec{x}+\tan{x}}.(\sec{x}\tan{x}+\sec^2{x}).e^xdx}+\int{e^x\sec{x}dx}\)
\(=e^x\ln{|\sec{x}+\tan{x}|}-\int{\frac{1}{\sec{x}+\tan{x}}.(\tan{x}+\sec{x})\sec{x}.e^xdx}+\int{e^x\sec{x}dx}\)
\(=e^x\ln{|\sec{x}+\tan{x}|}-\int{\sec{x}.e^xdx}+\int{e^x\sec{x}dx}\)
\(=e^x\ln{|\sec{x}+\tan{x}|}-\int{e^x\sec{x}dx}+\int{e^x\sec{x}dx}\)
\(=e^x\ln{|\sec{x}+\tan{x}|}+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।

\(\int{e^x\left\{(\ln{|x|})^2+\frac{2}{x}\ln{|x|}\right\}dx}\)
\(=\int{\left\{e^x(\ln{|x|})^2+e^x\frac{2}{x}\ln{|x|}\right\}dx}\)
\(=\int{e^x(\ln{|x|})^2dx}+\int{e^x\frac{2}{x}\ln{|x|}dx}\)
\(=(\ln{|x|})^2\int{e^xdx}-\int{\left\{\frac{d}{dx}(\ln{|x|})^2\int{e^xdx}\right\}dx}+\int{\frac{2}{x}e^x\ln{|x|}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=(\ln{|x|})^2.e^x-\int{2\ln{|x|}\frac{d}{dx}(\ln{|x|}).e^xdx}+\int{\frac{2}{x}e^x\ln{|x|}dx}\) ➜Note \(\because \int{e^xdx}=e^{x}, \frac{d}{dx}(x^2)=2x\)
\(=e^x(\ln{|x|})^2-\int{2\ln{|x|}.\frac{1}{x}.e^xdx}+\int{\frac{2}{x}e^x\ln{|x|}dx}\) ➜Note \(\because \frac{d}{dx}(\ln{|x|})=\frac{1}{x}\)
\(=e^x(\ln{|x|})^2-\int{\frac{2}{x}e^x\ln{|x|}dx}+\int{\frac{2}{x}e^x\ln{|x|}dx}\)
\(=e^x(\ln{|x|})^2+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।

\(\int{e^x\left\{\tan^{-1}{x}+\frac{1}{1+x^2}\right\}dx}\)
\(=\int{\left\{e^x\tan^{-1}{x}+e^x\frac{1}{1+x^2}\right\}dx}\)
\(=\tan^{-1}{x}\int{e^xdx}-\int{\left\{\frac{d}{dx}(\tan^{-1}{x})\int{e^xdx}\right\}dx}+\int{e^x\frac{1}{1+x^2}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\tan^{-1}{x}.e^x-\int{\frac{1}{1+x^2}.e^xdx}+\int{e^x\frac{1}{1+x^2}dx}\) ➜Note \(\because \int{e^xdx}=e^{x}, \frac{d}{dx}(\tan^{-1}{x})=\frac{1}{1+x^2}\)
\(=e^x\tan^{-1}{x}-\int{e^x\frac{1}{1+x^2}dx}+\int{e^x\frac{1}{1+x^2}dx}\)
\(=e^x\tan^{-1}{x}+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।

\(\int{e^x\left\{\frac{1}{\ln{|x|}}-\frac{1}{x(\ln{x})^2}\right\}dx}\)
\(=\int{\left\{e^x\frac{1}{\ln{|x|}}-e^x\frac{1}{x(\ln{x})^2}\right\}dx}\)
\(=\int{e^x\frac{1}{\ln{|x|}}dx}-\int{e^x\frac{1}{x(\ln{x})^2}dx}\)
\(=\frac{1}{\ln{|x|}}\int{e^xdx}-\int{\left\{\frac{d}{dx}\left(\frac{1}{\ln{|x|}}\right)\int{e^xdx}\right\}dx}-\int{e^x\frac{1}{x(\ln{x})^2}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{1}{\ln{|x|}}.e^x-\int{\left\{-\frac{1}{(\ln{|x|})^2}\right\}\frac{d}{dx}(\ln{|x|})e^xdx}-\int{e^x\frac{1}{x(\ln{x})^2}dx}\) ➜Note \(\because \int{e^xdx}=e^{x}, \frac{d}{dx}(\frac{1}{x})=-\frac{1}{x^2}\)
\(=\frac{e^x}{\ln{|x|}}+\int{\frac{1}{(\ln{|x|})^2}.\frac{1}{x}.e^xdx}-\int{e^x\frac{1}{x(\ln{x})^2}dx}\) ➜Note \(\because \int{e^xdx}=e^{x}, \frac{d}{dx}(\ln{|x|})=\frac{1}{x}\)
\(=\frac{e^x}{\ln{|x|}}+\int{e^x\frac{1}{x(\ln{x})^2}dx}-\int{e^x\frac{1}{x(\ln{x})^2}dx}\)
\(=\frac{e^x}{\ln{|x|}}+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।

\(\int{e^x\left(\frac{2+\sin{2x}}{1+\cos{2x}}\right)dx}\)
\(=\int{e^x\left(\frac{2}{1+\cos{2x}}+\frac{\sin{2x}}{1+\cos{2x}}\right)dx}\)
\(=\int{e^x\left(\frac{2}{2\cos^2{x}}+\frac{2\sin{x}\cos{x}}{2\cos^2{x}}\right)dx}\) ➜Note \(\because 1+\cos{2A}=2\cos^2{A}, \sin{2A}=2\sin{A}\cos{A}\)
\(=\int{e^x\left(\frac{1}{\cos^2{x}}+\frac{\sin{x}}{\cos{x}}\right)dx}\)
\(=\int{e^x\left(\sec^2{x}+\tan{x}\right)dx}\) ➜Note \(\because \frac{1}{\cos^2{A}}=\sec^2{A}, \frac{\sin{A}}{\cos{A}}=\tan{A}\)
\(=\int{\left(e^x\sec^2{x}+e^x\tan{x}\right)dx}\)
\(=\int{e^x\sec^2{x}dx}+\int{e^x\tan{x}dx}\)
\(=\int{e^x\sec^2{x}dx}+\tan{x}\int{e^xdx}-\int{\left\{\frac{d}{dx}(\tan{x})\int{e^xdx}\right\}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\int{e^x\sec^2{x}dx}+\tan{x}.e^x-\int{\sec^2{x}.e^xdx}\) ➜Note \(\because \int{e^xdx}=e^{x}, \frac{d}{dx}(\tan{x})=\sec^2{x}\)
\(=\int{e^x\sec^2{x}dx}+e^x\tan{x}-\int{e^x\sec^2{x}dx}\)
\(=e^x\tan{x}+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।

\(\int{e^x\left(\frac{1+\sin{x}}{1+\cos{x}}\right)dx}\)
\(=\int{e^x\left(\frac{1}{1+\cos{x}}+\frac{\sin{x}}{1+\cos{x}}\right)dx}\)
\(=\int{e^x\left\{\frac{1}{2\cos^2{\left(\frac{x}{2}\right)}}+\frac{2\sin{\left(\frac{x}{2}\right)}\cos{\left(\frac{x}{2}\right)}}{2\cos^2{\left(\frac{x}{2}\right)}}\right\}dx}\) ➜Note \(\because 1+\cos{A}=2\cos^2{\left(\frac{A}{2}\right)}, \sin{A}=2\sin{\left(\frac{A}{2}\right)}\cos{\left(\frac{A}{2}\right)}\)
\(=\int{e^x\left\{\frac{1}{2\cos^2{\left(\frac{x}{2}\right)}}+\frac{\sin{\left(\frac{x}{2}\right)}}{\cos{\left(\frac{x}{2}\right)}}\right\}dx}\)
\(=\int{e^x\left\{\frac{1}{2}\sec^2{\left(\frac{x}{2}\right)}+\tan{\left(\frac{x}{2}\right)}\right\}dx}\) ➜Note \(\because \frac{1}{\cos^2{\left(\frac{A}{2}\right)}}=\sec^2{\left(\frac{A}{2}\right)}, \frac{\sin{\left(\frac{A}{2}\right)}}{\cos{\left(\frac{A}{2}\right)}}=\tan{\left(\frac{A}{2}\right)}\)
\(=\int{\left\{\frac{1}{2}e^x\sec^2{\left(\frac{x}{2}\right)}+e^x\tan{\left(\frac{x}{2}\right)}\right\}dx}\)
\(=\frac{1}{2}\int{e^x\sec^2{\left(\frac{x}{2}\right)}dx}+\int{e^x\tan{\left(\frac{x}{2}\right)}dx}\)
\(=\frac{1}{2}\int{e^x\sec^2{\left(\frac{x}{2}\right)}dx}+\tan{\left(\frac{x}{2}\right)}\int{e^xdx}-\int{\left[\frac{d}{dx}\left\{\tan{\left(\frac{x}{2}\right)}\right\}\int{e^xdx}\right]dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{1}{2}\int{e^x\sec^2{\left(\frac{x}{2}\right)}dx}+\tan{\left(\frac{x}{2}\right)}.e^x-\int{\sec^2{\left(\frac{x}{2}\right)}.\frac{d}{dx}\left(\frac{x}{2}\right).e^xdx}\) ➜Note \(\because \int{e^xdx}=e^{x}, \frac{d}{dx}(\tan{x})=\sec^2{x}\)
\(=\frac{1}{2}\int{e^x\sec^2{\left(\frac{x}{2}\right)}dx}+\tan{\left(\frac{x}{2}\right)}.e^x-\int{\sec^2{\left(\frac{x}{2}\right)}.\frac{1}{2}.e^xdx}\) ➜Note \(\because \frac{d}{dx}\left(\frac{x}{2}\right)=\frac{1}{2}\)
\(=\frac{1}{2}\int{e^x\sec^2{\left(\frac{x}{2}\right)}dx}+e^x\tan{\left(\frac{x}{2}\right)}-\frac{1}{2}\int{e^x\sec^2{\left(\frac{x}{2}\right)}dx}\)
\(=e^x\tan{\left(\frac{x}{2}\right)}+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।

\(\int{e^{2x}\cos{\left(e^{x}\right)}dx}\)
\(=\int{e^{x}.e^{x}\cos{\left(e^{x}\right)}dx}\)
\(=\int{e^{x}\cos{\left(e^{x}\right)}e^{x}dx}\)
\(=\int{t\cos{t}dt}\)
\(=t\int{\cos{t}dt}-\int{\left\{\frac{d}{dt}(t)\int{\cos{t}dt}\right\}dt}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=t\sin{t}-\int{1.\sin{t}dt}\) ➜Note \(\because \int{\cos{x}dx}=\sin{x}, \frac{d}{dx}(x)=1\)
\(=t\sin{t}-\int{\sin{t}dt}\)
\(=t\sin{t}+\cos{t}+c\) ➜Note\(\because \int{\sin{x}dx}=-\cos{x}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=e^{x}\sin{\left(e^{x}\right)}+\cos{\left(e^{x}\right)}+c\) ➜Note\(\because t=e^{x}\)

\(\int{e^{x}\frac{x+1}{(x+2)^2}dx}\)
\(=\int{e^{x}\frac{x+2-1}{(x+2)^2}dx}\)
\(=\int{e^{x}\left\{\frac{x+2}{(x+2)^2}-\frac{1}{(x+2)^2}\right\}dx}\)
\(=\int{e^{x}\left\{\frac{1}{x+2}-\frac{1}{(x+2)^2}\right\}dx}\)
\(=\int{e^{x}\frac{1}{x+2}dx}-\int{e^{x}\frac{1}{(x+2)^2}dx}\)
\(=\frac{1}{x+2}\int{e^{x}dx}-\int{\left\{\frac{d}{dx}\left(\frac{1}{x+2}\right)\int{e^xdx}\right\}dx}-\int{e^{x}\frac{1}{(x+2)^2}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\frac{1}{x+2}e^{x}-\int{\left\{-\frac{1}{(x+2)^2}\right\}.e^xdx}-\int{e^{x}\frac{1}{(x+2)^2}dx}\) ➜Note \(\because \int{e^{x}dx}=e^{x}, \frac{d}{dx}\left(\frac{1}{x}\right)=-\frac{1}{x^2}\)
\(=\frac{1}{x+2}e^{x}+\int{e^x\frac{1}{(x+2)^2}dx}-\int{e^{x}\frac{1}{(x+2)^2}dx}\)
\(=\frac{1}{x+2}e^{x}+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{e^{x}}{x+2}+c\)

\(\int{x^n\ln{|x|}dx}\)
\(=\int{\ln{|x|}x^ndx}\)
\(=\ln{|x|}\int{x^ndx}-\int{\left\{\frac{d}{dx}(\ln{|x|})\int{x^ndx}\right\}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\ln{|x|}\frac{x^{n+1}}{n+1}-\int{\frac{1}{x}.\frac{x^{n+1}}{n+1}dx}\) ➜Note \(\because \int{x^ndx}=\frac{x^{n+1}}{n+1}, \frac{d}{dx}(\ln{|x|})=\frac{1}{x}\)
\(=\frac{x^{n+1}\ln{|x|}}{n+1}-\frac{1}{n+1}\int{x^{n+1-1}dx}\)
\(=\frac{x^{n+1}\ln{|x|}}{n+1}-\frac{1}{n+1}\int{x^{n}dx}\)
\(=\frac{x^{n+1}\ln{|x|}}{n+1}-\frac{1}{n+1}.\frac{x^{n+1}}{n+1}+c\) ➜Note\(\because \int{x^ndx}=\frac{x^{n+1}}{n+1}\) এবং \(c\) যোগজীকরণ ধ্রুবক।
\(=\frac{x^{n+1}\ln{|x|}}{n+1}-\frac{x^{n+1}}{(n+1)^2}+c\)

\(\int{e^{x}\sec{x}(1+\tan{x})dx}\)
\(=\int{e^{x}\sec{x}dx}+\int{e^x\sec{x}\tan{x}dx}\)
\(=\int{\sec{x}e^{x}dx}+\int{e^x\sec{x}\tan{x}dx}\)
\(=\sec{x}\int{e^{x}dx}-\int{\left\{\frac{d}{dx}(\sec{x})\int{e^xdx}\right\}dx}+\int{e^x\sec{x}\tan{x}dx}\) ➜Note \(\because \int{(uv)dx}=u\int{vdx}-\int{\left\{\frac{d}{dx}(u)\int{vdx}\right\}dx}\)
\(=\sec{x}.e^{x}-\int{\sec{x}\tan{x}.e^xdx}+\int{e^x\sec{x}\tan{x}dx}\) ➜Note \(\because \int{e^xdx}=e^x, \frac{d}{dx}(\sec{x})=\sec{x}\tan{x}\)
\(=e^x\sec{x}-\int{e^x\sec{x}\tan{x}dx}+\int{e^x\sec{x}\tan{x}dx}\)
\(=e^x\sec{x}+c\) ➜Note \(c\) যোগজীকরণ ধ্রুবক।