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

অনুশীলনী \(10.C / Q.2\)-এর প্রশ্নসমুহ
যোজিত ফল নির্ণয় করঃ
\(\int{\sin^m{x}\cos^n{x}dx}\), \(\int{\sin^m{x}dx}\), \(\int{\cos^n{x}dx}\) ও \(\int{e^{f(x)}f^{\prime}(x)dx}\) আকার।
\(Q.2.(i)\) \(\int{\sin^3{x}\cos{x}dx}\)
উত্তরঃ \(\frac{1}{4}\sin^4{x}+c\)

\(Q.2.(ii)\) \(\int{\sin^2{x}\cos{x}dx}\)
উত্তরঃ \(\frac{1}{3}\sin^3{x}+c\)
[ ঢাঃ২০০২]

\(Q.2.(iii)\) \(\int{\sin^2{\frac{x}{2}}\cos{\frac{x}{2}}dx}\)
উত্তরঃ \(\frac{2}{3}\sin^3{\frac{x}{2}}+c\)
[ কুঃ২০০৩ ]

\(Q.2.(iv)\) \(\int{\sin^3{x}\cos^4{x}dx}\)
উত্তরঃ \(\frac{1}{7}\cos^7{x}-\frac{1}{5}\cos^5{x}+c\)
[ রাঃ২০০১ ]

\(Q.2.(v)\) \(\int{\sin^3{x}\cos^3{x}dx}\)
উত্তরঃ \(\frac{1}{4}\sin^4{x}-\frac{1}{6}\sin^6{x}+c\)
[ যঃ২০০৬ ]

\(Q.2.(vi)\) \(\int{\sin^5{x}dx}\)
উত্তরঃ \(-\cos{x}+\frac{2}{3}\cos^3{x}-\frac{1}{5}\cos^5{x}+c\)
[ রুয়েটঃ২০১০-২০১১ ]

\(Q.2.(vii)\) \(\int{xe^{x^2}dx}\)
উত্তরঃ \(\frac{1}{2}e^{x^2}+c\)
[ বঃ২০০৩ ]

\(Q.2.(viii)\) \(\int{x^2a^{x^3}dx}\)
উত্তরঃ \(\frac{a^{x^3}}{3\ln{a}}+c\)
[ মাঃ২০০৯ ]

\(Q.2.(ix)\) \(\int{\frac{e^{\sqrt{x}}}{5\sqrt{x}}dx}\)
উত্তরঃ \(\frac{2}{5}e^{\sqrt{x}}+c\)

\(Q.2.(x)\) \(\int{\left(1-\frac{1}{x^2}\right)e^{x+\frac{1}{x}}dx}\)
উত্তরঃ \(e^{x+\frac{1}{x}}+c\)
[ বিআইটিঃ ২০০১-২০০২]

\(Q.2.(xi)\) \(\int{\cos{x}e^{\sin{x}}dx}\)
উত্তরঃ \(e^{\sin{x}}+c\)
[ ঢাঃ ২০১১; রাঃ২০০৮ ]

\(Q.2.(xii)\) \(\int{\sec^2{x}e^{\tan{x}}dx}\)
উত্তরঃ \(e^{\tan{x}}+c\)

\(Q.2.(xiii)\) \(\int{\frac{e^{\sin^{-1}{x}}}{\sqrt{1-x^2}}dx}\)
উত্তরঃ \(e^{\sin^{-1}{x}}+c\)
[ বুয়েটঃ২০০৬; বুটেক্সঃ২০১০-২০১১; কুয়েটঃ২০০৬-২০০৭; চঃ২০১১ ]
\(Q.2.(xiv)\) \(\int{e^{a\sin^{-1}{x}}.\frac{dx}{\sqrt{1-x^2}}}\)
উত্তরঃ \(\frac{1}{a}e^{a\sin^{-1}{x}}+c\)

\(Q.2.(xv)\) \(\int{e^{\tan^{-1}{x}}.\frac{dx}{1+x^2}}\)
উত্তরঃ \(e^{\tan^{-1}{x}}+c\)
[ ঢাঃ২০০৯; মাঃ২০১২,২০১৪ ]

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

\(Q.2.(xvii)\) \(\int{\sin^5{\theta}\cos{\theta}d\theta}\)
উত্তরঃ \(\frac{1}{6}\sin^6{\theta}+c\)

\(Q.2.(xviii)\) \(\int{\cos^3{x}\sqrt{\sin{x}}dx}\)
উত্তরঃ \(\frac{2}{3}\sin^{\frac{3}{2}}{x}-\frac{2}{7}\sin^{\frac{7}{2}}{x}+c\)

\(Q.2.(xix)\) \(\int{\sin^2{x}\cos^2{x}dx}\)
উত্তরঃ \(\frac{1}{8}x-\frac{1}{8}\sin{4x}+c\)

\(Q.2.(xx)\) \(\int{\sin^2{x}\cos{2x}dx}\)
উত্তরঃ \(\frac{1}{16}(4\sin{2x}-4x-\sin{4x})+c\)

\(Q.2.(xxi)\) \(\int{\sin^4{x}\cos^4{x}dx}\)
উত্তরঃ \(\frac{1}{1024}(24x-8\sin{4x}+\sin{8x})+c\)

\(Q.2.(xxii)\) \(\int{\frac{\sin{4x}}{\sin^4{x}+\cos^4{x}}dx}\)
উত্তরঃ \(-ln{|\sin^4{x}+\cos^4{x}|}+c\)

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

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

\(Q.2.(xxv)\) \(\int{(\sqrt{\tan{x}}+\sqrt{\cot{x}})dx}\)
উত্তরঃ \(\sqrt{2}\sin^{-1}{\sin{x}-\cos{x}}+c\)

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

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