বিশেষ আকারের যোগজীকরণ-১ (Integration of spacial type-1)

অনুশীলনী \(10.C / Q.1\)-এর প্রশ্নসমুহ
যোজিত ফল নির্ণয় করঃ
\(\int{f\{g(x)\}g^{\prime}(x)dx}\) ও \(\int{\{f(x)\}^nf^{\prime}(x)dx}\) আকার।
\(Q.1.(i)\) \(\int{x\sin{x^2}dx}\)
উত্তরঃ \(-\frac{1}{2}\cos{x^2}+c\)

\(Q.1.(ii)\) \(\int{x^2\cos{x^3}dx}\)
উত্তরঃ \(\frac{1}{3}\sin{x^3}+c\)

\(Q.1.(iii)\) \(\int{\frac{\sin{\frac{1}{x}}}{x^2}dx}\)
উত্তরঃ \(\cos{\frac{1}{x}}+c\)
[ যঃ২০০৭; ঢাঃ২০০৪ ]

\(Q.1.(iv)\) \(\int{\frac{\sec{\sqrt{x}}}{\sqrt{x}}dx}\)
উত্তরঃ \(2\ln{|\sec{\sqrt{x}}+\tan{\sqrt{x}}|}+c\)

\(Q.1.(v)\) \(\int{\frac{\cos{\sqrt{x}}}{\sqrt{x}}dx}\)
উত্তরঃ \(2\sin{\sqrt{x}}+c\)

\(Q.1.(vi)\) \(\int{e^x\cos{e^x}dx}\)
উত্তরঃ \(\sin{e^x}+c\)

\(Q.1.(vii)\) \(\int{e^x\tan{e^x}\sec{e^x}dx}\)
উত্তরঃ \(\sec{e^x}+c\)
[ কুঃ২০০৩ ]

\(Q.1.(viii)\) \(\int{e^{2x}\tan{e^{2x}}\sec{e^{2x}}dx}\)
উত্তরঃ \(\frac{1}{2}\sec{e^{2x}}+c\)
[ চঃ২০০৭ ]

\(Q.1.(ix)\) \(\int{\frac{e^{x}(1+x)dx}{\cos^2{(xe^x)}}}\)
উত্তরঃ \(\tan{(xe^{x})}+c\)
[ রুয়েটঃ২০১১-২০১২]

\(Q.1.(x)\) \(\int{\frac{\cos{(\ln{|x|})}}{x}dx}\)
উত্তরঃ \(\sin{(\ln{|x|})}+c\)

\(Q.1.(xi)\) \(\int{\frac{\tan^2{(\ln{|x|})}}{x}dx}\)
উত্তরঃ \(\tan{(\ln{|x|})}-\ln{|x|}+c\)
[ বঃ২০০২ ]

\(Q.1.(xii)\) \(\int{\frac{\tan{(\ln{|x|})}}{x}dx}\)
উত্তরঃ \(\ln{\{\sec{(\ln{|x|})}\}}+c\)

\(Q.1.(xiii)\) \(\int{\frac{\tan{(\sin^{-1}{x})}}{\sqrt{1-x^2}}dx}\)
উত্তরঃ \(\ln{\{|\sec{(\sin^{-1}{x})}|\}}+c\)
[ বঃ২০১১,২০০৭; যঃ২০০৯; কুঃ২০০৭,২০০৪; সিঃ২০০৬; রাঃ২০০৫; ঢাঃ২০০৩; মাঃ২০১১,২০০৭ ]

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

\(Q.1.(xv)\) \(\int{x^2\sqrt(1-x^3)dx}\)
উত্তরঃ \(-\frac{2}{9}(1-x^3)^{\frac{3}{2}}+c\)

\(Q.1.(xvi)\) \(\int{\frac{e^{-x}dx}{(5+e^{-x})^2}}\)
উত্তরঃ \(\frac{1}{5+e^{-x}}+c\)

\(Q.1.(xvii)\) \(\int{\left(e^x+\frac{1}{x}\right)(e^x+\ln{|x|})dx}\)
উত্তরঃ \(\frac{1}{2}(e^x+\ln{|x|})^2+c\)

\(Q.1.(xviii)\) \(\int{\frac{(\ln{|x|})^2}{x}dx}\)
উত্তরঃ \(\frac{1}{3}(\ln{|x|})^3+c\)

\(Q.1.(xix)\) \(\int{\frac{\ln{|x|}}{x}dx}\)
উত্তরঃ \(\frac{1}{2}(\ln{|x|})^2+c\)

\(Q.1.(xx)\) \(\int{\frac{1}{x(\ln{|x|})^2}dx}\)
উত্তরঃ \(-\frac{1}{\ln{|x|}}+c\)
\(Q.1.(xxi)\) \(\int{\frac{\sqrt{1+\ln{|x|}}}{x}dx}\)
উত্তরঃ \(\frac{2}{3}(1+\ln{|x|})^{\frac{3}{2}}+c\)

\(Q.1.(xxii)\) \(\int{\frac{1}{x(1+\ln{|x|})^3}dx}\)
উত্তরঃ \(-\frac{1}{2(1+\ln{|x|})^2}+c\)
[ ঢাঃ২০০১]

\(Q.1.(xxiii)\) \(\int{(1+\cos{x})^3\sin{x}dx}\)
উত্তরঃ \(-\frac{1}{4}(1+\cos{x})^4+c\)
[ কুঃ২০০৩]

\(Q.1.(xxiv)\) \(\int{\sqrt{1-\sin{x}}\cos{x}dx}\)
উত্তরঃ \(-\frac{2}{3}(1-\sin{x})^{\frac{3}{2}}+c\)
[ সিঃ২০০১]

\(Q.1.(xxv)\) \(\int{\frac{\cos{x}dx}{(1-\sin{x})^2}}\)
উত্তরঃ \(\frac{1}{1-\sin{x}}+c\)
[ কুঃ২০০৬; বঃ২০১১ ]

\(Q.1.(xxvi)\) \(\int{\tan^3{x}\sec^2{x}dx}\)
উত্তরঃ \(\frac{1}{4}\tan^4{x}+c\)

\(Q.1.(xxvii)\) \(\int{\frac{1+\tan^2{x}}{(1+\tan{x})^2}dx}\)
উত্তরঃ \(-\frac{1}{1+\tan{x}}+c\)
[ কুয়েটঃ ২০১৩-২০১৪ ]

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

\(Q.1.(xxix)\) \(\int{\frac{2x\sin^{-1}{x^2}}{\sqrt{1-x^4}}dx}\)
উত্তরঃ \(\frac{1}{2}(\sin^{-1}{x^2})^2+c\)

\(Q.1.(xxx)\) \(\int{\frac{x^2\tan^{-1}{x^3}}{1+x^6}dx}\)
উত্তরঃ \(\frac{1}{6}(\tan^{-1}{x^3})^2+c\)
[ কুঃ২০১৫,২০০৮; যঃ২০০৬; চঃ২০০৭; মাঃ২০০৮ ]

\(Q.1.(xxxi)\) \(\int{x7^{x^2}dx}\)
উত্তরঃ \(\frac{7^{x^2}}{2\ln{7}}+c\)

\(Q.1.(xxxii)\) \(\int{\frac{\tan{\sqrt{x}}\sec^2{\sqrt{x}}}{\sqrt{x}}dx}\)
উত্তরঃ \(\tan^2{\sqrt{x}}+c\)

\(Q.1.(xxxiii)\) \(\int{\tan^2{x}\sec^2{x}dx}\)
উত্তরঃ \(\frac{1}{3}\tan^3{x}+c\)

\(Q.1.(xxxiv)\) \(\int{\tan^4{x}\sec^2{x}dx}\)
উত্তরঃ \(\frac{1}{5}\tan^5{x}+c\)

\(Q.1.(xxxv)\) \(\int{\cos{x}\cos{(\sin{x})}dx}\)
উত্তরঃ \(\sin{(\sin{x})}+c\)

\(Q.1.(xxxvi)\) \(\int{\sin{x}\sin{(\cos{x})}dx}\)
উত্তরঃ \(\cos{(\cos{x})}+c\)

\(Q.1.(xxxvii)\) \(\int{\frac{(\sec^{-1}{x})^4}{x\sqrt{x^2-1}}dx}\)
উত্তরঃ \(\frac{1}{5}(\sec^{-1}{x})^5+c\)

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

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

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