অংশায়ন পদ্ধতিতে যোগজীকরণ (Integration by parts)

অনুশীলনী \(10.F / Q.3\)-এর প্রশ্নসমুহ

যোজিত ফল নির্ণয় করঃ
\(Q.3.(i)\) \(\int{x^3e^{x^2}dx}\)
উত্তরঃ \(\frac{1}{2}e^{x^2}(x^2-1)+c\)

\(Q.3.(ii)\) \(\int{e^{2t}\cos{\left(e^{t}\right)}dt}\)
উত্তরঃ \(e^{t}\sin{\left(e^{t}\right)}+\cos{\left(e^{t}\right)}+c\)

\(Q.3.(iii)\) \(\int{x\sin^{-1}{(x^2)}dx}\)
উত্তরঃ \(\frac{1}{2}\left\{x^2\sin^{-1}{(x^2)}+\sqrt{1-x^4}\right\}+c\)
[ রাঃ২০০৬; ঢাঃ২০০৫ ]

\(Q.3.(iv)\) \(\int{\frac{\ln{|\sec^{-1}{x}|}}{x\sqrt{x^2-1}}dx}\)
উত্তরঃ \(\sec^{-1}{x}(\ln{|\sec^{-1}{x}|}-1)+c\)
[ ঢাঃ২০০৮ ]

\(Q.3.(v)\) \(\int{e^{x}(\sin{x}+\cos{x})dx}\)
উত্তরঃ \(e^{x}\sin{x}+c\)
[ সিঃ২০১১,২০০৫; ঢাঃ২০১০; বঃ২০০৬; কুঃ২০০০ ]

\(Q.3.(vi)\) \(\int{\frac{e^{x}}{x}(1+x\ln{|x|})dx}\)
উত্তরঃ \(e^{x}\ln{|x|}+c\)
[ যঃ২০০৭; বঃ২০০১; কুঃ২০০০ ]

\(Q.3.(vii)\) \(\int{e^{x}\left\{\frac{1}{1-x}+\frac{1}{(1-x)^2}\right\}dx}\)
উত্তরঃ \(\frac{e^{x}}{1-x}+c\)

\(Q.3.(viii)\) \(\int{e^{5x}\left\{5\ln{|x|}+\frac{1}{x}\right\}dx}\)
উত্তরঃ \(e^{5x}\ln{|x|}+c\)
[ চঃ২০০৯ ]

\(Q.3.(ix)\) \(\int{e^{-x}\left\{\frac{1}{x}+\frac{1}{x^2}\right\}dx}\)
উত্তরঃ \(-\frac{e^{-x}}{x}+c\)

\(Q.3.(x)\) \(\int{\frac{xe^{x}}{(x+1)^2}dx}\)
উত্তরঃ \(\frac{e^{x}}{x+1}+c\)
[ ঢাঃ২০১১; কুঃ২০১১,২০০৯; চঃ২০১১,২০০৫,২০০৪ ]

\(Q.3.(xi)\) \(\int{e^{x}(\tan{x}-\ln{|\cos{x}|})dx}\)
উত্তরঃ \(-e^{x}\ln{|\cos{x}|}+c\)
[ কুঃ২০০৩ ]

\(Q.3.(xii)\) \(\int{cosec^3{x}dx}\)
উত্তরঃ \(\frac{1}{2}\left\{\ln{\left|\tan{\left(\frac{x}{2}\right)}\right|}-cosec \ {x}\cot{x}\right\}+c\)
[ কুঃ২০০২ ]

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

\(Q.3.(xiv)\) \(\int{\frac{e^{x}(x^2+1)}{(x+1)^2}dx}\)
উত্তরঃ \(\frac{e^{x}(x-1)}{x+1}+c\)
[ কুয়েটঃ২০০৪-২০০৫ ]
\(Q.3.(xv)\) \(\int{x\sec^2{3x}dx}\)
উত্তরঃ \(\frac{1}{9}(3x\tan{3x}-\ln{|\sec{3x}|})+c\)

\(Q.3.(xvi)\) \(\int{x\tan^2{x}dx}\)
উত্তরঃ \(x\tan{x}-\ln{|\sec{x}|}-\frac{1}{2}x^2+c\)
[ রাঃ, সিঃ ২০০৫ ]

\(Q.3.(xvii)\) \(\int{e^{-3x}\cos{4x}dx}\)
উত্তরঃ \(\frac{1}{25}e^{-3x}(4\sin{4x}-3\cos{4x})+c\)

\(Q.3.(xviii)\) \(\int{\log_{10}{x}dx}\)
উত্তরঃ \(x\log_{10}{x}-x\log_{10}{e}+c\)

\(Q.3.(xix)\) \(\int{e^{-2x}\left(\frac{1}{x}-2\ln{|x|}\right)dx}\)
উত্তরঃ \(e^{-2x}\ln{|x|}+c\)

\(Q.3.(xx)\) \(\int{\{\sin{(\ln{|x|})}+\cos{(\ln{|x|})}\}dx}\)
উত্তরঃ \(x\sin{(\ln{|x|})}+c\)

\(Q.3.(xxi)\) \(\int{e^x\{\ln{|\sec{x}+\tan{x}|}+\sec{x}\}dx}\)
উত্তরঃ \(e^x\ln{|\sec{x}+\tan{x}|}+c\)

\(Q.3.(xxii)\) \(\int{e^x\left\{(\ln{|x|})^2+\frac{2}{x}\ln{|x|}\right\}dx}\)
উত্তরঃ \(e^x(\ln{|x|})^2+c\)

\(Q.3.(xxiii)\) \(\int{e^x\left\{\tan^{-1}{x}+\frac{1}{1+x^2}\right\}dx}\)
উত্তরঃ \(e^x\tan^{-1}{x}+c\)

\(Q.3.(xxiv)\) \(\int{e^x\left\{\frac{1}{\ln{|x|}}-\frac{1}{x(\ln{x})^2}\right\}dx}\)
উত্তরঃ \(\frac{e^x}{\ln{|x|}}+c\)

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

\(Q.3.(xxvi)\) \(\int{e^x\left(\frac{1+\sin{x}}{1+\cos{x}}\right)dx}\)
উত্তরঃ \(e^x\tan{\left(\frac{x}{2}\right)}+c\)
[ বুয়েটঃ ২০১-২০১৩]

\(Q.3.(xxvii)\) \(\int{e^{2x}\cos{\left(e^{x}\right)}dx}\)
উত্তরঃ \(e^{x}\sin{\left(e^{x}\right)}+\cos{\left(e^{x}\right)}+c\)

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

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

\(Q.3.(xxx)\) \(\int{e^{x}\sec{x}(1+\tan{x})dx}\)
উত্তরঃ \(e^{x}\sec{x}+c\)
[ যঃ২০১১; রাঃ২০০৩ ]
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