বিশেষ আকারের যোগজীকরণ-২ (Integration of spacial type-2)

অনুশীলনী \(10.D / Q.3\)-এর প্রশ্নসমুহ

যোজিত ফল নির্ণয় করঃ
\(Q.3.(i)\) \(\int{\frac{dx}{(x+2)\sqrt{x+3}}}\)
উত্তরঃ \(\ln{\left|\frac{\sqrt{x+3}-1}{\sqrt{x+3}+1}\right|}+c\)

\(Q.3.(ii)\) \(\int{\frac{dx}{(2x+1)\sqrt{4x+3}}}\)
উত্তরঃ \(\frac{1}{2}\ln{\left|\frac{\sqrt{4x+3}-1}{\sqrt{4x+3}+1}\right|}+c\)

\(Q.3.(iii)\) \(\int{\frac{dx}{(2x+1)\sqrt{x+3}}}\)
উত্তরঃ \(\frac{1}{\sqrt{10}}\ln{\left|\frac{\sqrt{2x+6}-\sqrt{5}}{\sqrt{2x+6}+\sqrt{5}}\right|}+c\)

\(Q.3.(iv)\) \(\int{\frac{dx}{x^{\frac{1}{2}}-x^{\frac{1}{4}}}}\)
উত্তরঃ \(2\sqrt{x}+4\sqrt[4]{x}+4\ln{|\sqrt[4]{x}-1|}+c\)
[ চঃ২০১০,২০১৫; রাঃ২০০৭; যঃ২০০০ ]

\(Q.3.(v)\) \(\int{\frac{\sqrt{x}dx}{1+\sqrt[3]{x}}}\) এর যোজিত ফল নির্ণয় কর।
উত্তরঃ \(\frac{6}{7}x^{\frac{7}{6}}-\frac{6}{5}x^{\frac{5}{6}}+2x^{\frac{1}{2}}\) \(-6x^{\frac{1}{6}}+6\tan^{-1}{\left(x^{\frac{1}{6}}\right)}+c\)
[ চঃ২০০০ ]

\(Q.3.(vi)\) \(\int{\frac{1}{1+\tan{x}}dx}\)
উত্তরঃ \(\frac{1}{2}(x+\ln{|\cos{x}+\sin{x}|})+c\)
[ দিঃ২০১৪; রাঃ২০১২,২০০৮; যঃ২০১২; বঃ২০১৪,২০১২,২০০৯; ঢাঃ২০১০,২০০৮ ]

\(Q.3.(vii)\) \(\int{\frac{dx}{2+\cot{x}}}\)
উত্তরঃ \(\frac{1}{5}(2x-\ln{|2\sin{x}+\cos{x}|})+c\)

\(Q.3.(viii)\) \(\int{\frac{3\sin{x}dx}{\cos{x}+\sin{x}}}\)
উত্তরঃ \(\frac{3}{2}(x-\ln{|\cos{x}+\sin{x}|})+c\)

\(Q.3.(ix)\) \(\int{\frac{dx}{a+b\cos{x}}}\)
উত্তরঃ \(\frac{2}{\sqrt{a^2-b^2}}\tan^{-1}{\left(\frac{\sqrt{a-b}\tan{\frac{x}{2}}}{\sqrt{a+b}}\right)}+c\)
[ রুয়েটঃ২০১০-২০১১ ]

\(Q.3.(x)\) \(\int{\frac{dx}{1+\sin{x}-\cos{x}}}\)
উত্তরঃ \(\ln{\left|\frac{\tan{\frac{x}{2}}}{1+\tan{\frac{x}{2}}}\right|}+c\)
[ বুয়েটঃ২০১১-২০১২ ]

\(Q.3.(xi)\) \(\int{\frac{dx}{a\cos{x}+b\sin{x}}}\)
উত্তরঃ \(\frac{1}{\sqrt{a^2+b^2}}\ln{\left|\tan{\frac{1}{2}\left\{x+\tan^{-1}{\left(\frac{a}{b}\right)}\right\}}\right|}+c\)
[ কুয়েটঃ২০০৯-২০১০ ]

\(Q.3.(xii)\) \(\int{\sqrt{\frac{1-x}{1+x}}dx}\)
উত্তরঃ \(\sin^{-1}{x}+\sqrt{1-x^2}+c\)

\(Q.3.(xiii)\) \(\int{\sqrt{\frac{1+x}{1-x}}dx}\)
উত্তরঃ \(\sin^{-1}{x}-\sqrt{1-x^2}+c\)
[ কুয়েটঃ২০১১-২০১২ ]

\(Q.3.(xiv)\) \(\int{\sqrt{\frac{5-x}{5+x}}dx}\)
উত্তরঃ \(5\sin^{-1}{\left(\frac{x}{5}\right)}+\sqrt{25-x^2}+c\)

\(Q.3.(xv)\) \(\int{\frac{dx}{\sqrt{(x-2)(x-3)}}}\)
উত্তরঃ \(2\ln{|\sqrt{x-2}+\sqrt{x-3}|}+c\)

\(Q.3.(xvi)\) \(\int{\frac{dx}{\sqrt{(x-2)(3-x)}}}\)
উত্তরঃ \(2\sin^{-1}{(\sqrt{x-2})}+c\)

\(Q.3.(xvii)\) \(\int{\sqrt{\frac{a+x}{x}}dx}\)
উত্তরঃ \(\sqrt{ax+x^2}+\frac{a}{2}\ln{\left|\sqrt{ax+x^2}+\left(x+\frac{a}{2}\right)\right|}+c\)

\(Q.3.(xviii)\) \(\int{\sqrt{\tan{y}}+\sqrt{\cot{y}}dy}\)
উত্তরঃ \(\sqrt{2}\sin^{-1}{(\sin{y}-\cos{y})}+c\)

\(Q.3.(xix)\) \(\int{\sqrt{\tan{y}}dy}\)
উত্তরঃ \(\frac{1}{\sqrt{2}}\tan^{-1}\left(\frac{\tan{y}-1}{\sqrt{2\tan{y}}}\right)\) \(+\frac{1}{2\sqrt{2}}\ln{\left|\frac{\tan{y}-\sqrt{2\tan{y}}+1}{\tan{y}+\sqrt{2\tan{y}}+1}\right|}+c\)

\(Q.3.(xx)\) \(\int{\frac{\sin{y}dy}{\sin{(y+a)}}}\)
উত্তরঃ \(y\cos{a}-\sin{a}\ln{|\sin{(y+a)}|}+c\)

\(Q.3.(xxi)\) \(\int{\frac{dy}{\sqrt{y}+\sqrt{1-y}}}\)
উত্তরঃ \(\sqrt{y}-\sqrt{1-y}-\frac{1}{\sqrt{2}}\ln{\left|\tan{\frac{1}{2}\left\{\sin^{-1}{(\sqrt{y})}+\frac{\pi}{4}\right\}}\right|}+c\)

\(Q.3.(xxii)\) \(\int{\frac{dx}{4+5\sin{x}}}\)
উত্তরঃ \(\frac{1}{3}\ln{\left|\frac{2\tan{\left(\frac{x}{2}\right)}+1}{2\tan{\left(\frac{x}{2}\right)}+4}\right|}+c\)

\(Q.3.(xxiii)\) \(\int{\frac{dx}{5+4\cos{x}}}\)
উত্তরঃ \(\frac{2}{3}\tan^{-1}{\left\{\frac{1}{3}\tan{\left(\frac{x}{2}\right)}\right\}}+c\)
\(Q.3.(xxiv)\) \(\int{\frac{dx}{3+2\cos{x}}}\)
উত্তরঃ \(\frac{2}{\sqrt{5}}\tan^{-1}{\left\{\frac{1}{\sqrt{5}}\tan{\left(\frac{x}{2}\right)}\right\}}+c\)

\(Q.3.(xxv)\) \(\int{\frac{dx}{4+5\cos{x}}}\)
উত্তরঃ \(\frac{1}{3}\ln{\left|\frac{3+\tan{\left(\frac{x}{2}\right)}}{3-\tan{\left(\frac{x}{2}\right)}}\right|}+c\)

\(Q.3.(xxvi)\) \(\int{\frac{dx}{5-13\sin{x}}}\)
উত্তরঃ \(\frac{1}{12}\ln{\left|\frac{5\tan{\left(\frac{x}{2}\right)}-25}{5\tan{\left(\frac{x}{2}\right)}-1}\right|}+c\)

\(Q.3.(xxvii)\) \(\int{\frac{dx}{5+4\sin{x}}}\)
উত্তরঃ \(\frac{2}{3}\tan^{-1}{\left[\frac{1}{3}\left\{5\tan{\left(\frac{x}{2}\right)}+4\right\}\right]}+c\)

\(Q.3.(xxviii)\) \(\int{\frac{dx}{3+5\cos{x}}}\)
উত্তরঃ \(\frac{1}{4}\ln{\left|\frac{2+\tan{\left(\frac{x}{2}\right)}}{2-\tan{\left(\frac{x}{2}\right)}}\right|}+c\)

\(Q.3.(xxix)\) \(\int{\frac{5\sin{x}+4\cos{x}}{2\sin{x}+3\cos{x}}dx}\)
উত্তরঃ \(\frac{22x}{13}-\frac{7}{13}\ln{\left|2\sin{x}+3\cos{x}\right|}+c\)

\(Q.3.(xxx)\) \(\int{\frac{4\sin{x}+3\cos{x}}{5\sin{x}+4\cos{x}}dx}\)
উত্তরঃ \(\frac{32x}{41}-\frac{1}{41}\ln{\left|5\sin{x}+4\cos{x}\right|}+c\)

\(Q.3.(xxxi)\) \(\int{\frac{11\cos{x}-16\sin{x}}{2\cos{x}+5\sin{x}}dx}\)
উত্তরঃ \(3\ln{\left|2\cos{x}+5\sin{x}\right|}-2x+c\)

\(Q.3.(xxxii)\) \(\int{\frac{2\sin{x}+3\cos{x}}{3\sin{x}+4\cos{x}}dx}\)
উত্তরঃ \(\frac{18x}{25}+\frac{1}{25}\ln{\left|3\sin{x}+4\cos{x}\right|}+c\)

\(Q.3.(xxxiii)\) \(\int{\frac{6+3\sin{x}+14\cos{x}}{3+4\sin{x}+5\cos{x}}dx}\)
উত্তরঃ \(2x+\ln{\left|3+4\sin{x}+5\cos{x}\right|}+c\)

\(Q.3.(xxxiv)\) \(\int{\frac{1-\sin{x}-\cos{x}}{1+\sin{x}-\cos{x}}dx}\)
উত্তরঃ \(\ln{\left|\frac{\tan{\left(\frac{x}{2}\right)}}{1+\tan{\left(\frac{x}{2}\right)}}\right|}-\ln{|1+\sin{x}-\cos{x}|}+c\)

\(Q.3.(xxxv)\) \(\int{\frac{\cos{x}+2\sin{x}+3}{4\cos{x}+5\sin{x}+6}dx}\)
উত্তরঃ \(\frac{14x}{41}-\frac{3}{41}\ln{\left|4\cos{x}+5\sin{x}+6\right|}\)
\(+\frac{39}{41\sqrt{5}}\ln{\left|\frac{2\tan{\left(\frac{x}{2}\right)}+5-\sqrt{5}}{2\tan{\left(\frac{x}{2}\right)}+5+\sqrt{5}}\right|}+c\)

\(Q.3.(xxxvi)\) \(\int{\frac{2\sin{x}+\cos{x}+3}{\sin{x}+2\cos{x}+4}dx}\)
উত্তরঃ \(\frac{4x}{5}-\frac{3}{5}\ln{\left|\sin{x}+2\cos{x}+4\right|}-\frac{2}{5\sqrt{11}}\tan^{-1}{\left[\frac{1}{\sqrt{11}}\left\{1+2\tan{\left(\frac{x}{2}\right)}\right\}\right]}+c\)

\(Q.3.(xxxvii)\) \(\int{\frac{dx}{(x-1)\sqrt{x+4}}}\)
উত্তরঃ \(\frac{1}{\sqrt{5}}\ln{\left|\frac{\sqrt{x+4}-\sqrt{5}}{\sqrt{x+4}+\sqrt{5}}\right|}+c\)

\(Q.3.(xxxviii)\) \(\int{\frac{dx}{(x-3)\sqrt{x+1}}}\)
উত্তরঃ \(\frac{1}{2}\ln{\left|\frac{\sqrt{x+1}-2}{\sqrt{x+1}+2}\right|}+c\)
[ ঢাঃ২০১০,বঃ২০১৩ ]

\(Q.3.(xxxix)\) \(\int{\frac{dx}{(1-x)\sqrt{1-x^2}}}\)
উত্তরঃ \(\sqrt{\frac{1+x}{1-x}}+c\)
[ ঢাঃ২০১০,বঃ২০১৩ ]

\(Q.3.(xL)\) \(\int{\sqrt{1+\sec{x}}dx}\)
উত্তরঃ \(2\sin^{-1}{\left\{\sqrt{2}\sin{\left(\frac{x}{2}\right)}\right\}}+c\)

\(Q.3.(xLi)\) \(\int{\cos{\left\{2\cot^{-1}\sqrt{\frac{1-x}{1+x}}\right\}}dx}\)
উত্তরঃ \(-\frac{1}{2}x^2+c\)

\(Q.3.(xLii)\) \(\int{\sin{\left\{2\tan^{-1}\sqrt{\frac{1-x}{1+x}}\right\}}dx}\)
উত্তরঃ \(-\frac{1}{2}\cos^{-1}{x}+\frac{1}{2}x\sqrt{1-x^2}+c\)

\(Q.3.(xLiii)\) \(\int{\frac{dx}{x+\sqrt{x}}}\)
উত্তরঃ \(2\ln{(\sqrt{x}+1)}+c\)

\(Q.3.(xLiv)\) \(\int{x\sqrt{\frac{1-x}{1+x}}dx}\)
উত্তরঃ \(\frac{x\sqrt{1-x^2}}{2}-\frac{1}{2}\sin^{-1}{x}-\sqrt{1-x^2}+c\)

\(Q.3.(xLv)\) \(\int{\sin^{-1}{\sqrt{\frac{x}{a+x}}}dx}\)
উত্তরঃ \((a+x)\tan^{-1}{\left(\sqrt{\frac{x}{a}}\right)}-\sqrt{(ax)}+c\)
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