যোগজীকরণ-প্রতিস্থাপন পদ্ধতি ( Inetgration- Replacement method)

অনুশীলনী \(10.B / Q.2\)-এর প্রশ্নসমুহ

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

\(Q.2.(ii)\) \(\int{\frac{dt}{2e^{2t}}}\)
উত্তরঃ \(-\frac{1}{4e^{2t}}+c\)

\(Q.2.(iii)\) \(\int{a^{4x}dx}\)
উত্তরঃ \(\frac{a^{4x}}{4\ln{a}}+c\)

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

\(Q.2.(v)\) \(\int{\frac{(e^x+1)^2}{\sqrt{e^x}}dx}\)
উত্তরঃ \(\frac{2}{3}e^{\frac{3x}{2}}+4e^{\frac{x}{2}}-2e^{-\frac{x}{2}}+c\)
[ চঃ ২০১৪; মাঃ ২০১০ ]

\(Q.2.(vi)\) \(\int{\frac{e^{5x}+e^{3x}}{e^x+e^{-x}}dx}\)
উত্তরঃ \(\frac{1}{4}e^{4x}+c\)
[ চঃ ২০০০; দিঃ ২০১০ ]

\(Q.2.(vii)\) \(\int{\sin{7x}dx}\)
উত্তরঃ \(-\frac{1}{7}\cos{7x}+c\)
[ সিঃ ২০০৫ ]

\(Q.2.(viii)\) \(\int{\cos{7x}dx}\)
উত্তরঃ \(\frac{1}{7}\sin{7x}+c\)

\(Q.2.(ix)\) \(\int{\sin{5x}dx}\)
উত্তরঃ \(-\frac{1}{5}\cos{5x}+c\)
[ মাঃ ২০১৫ ]

\(Q.2.(x)\) \(\int{\cos{\left(5x+\frac{\pi}{3}\right)}dx}\)
উত্তরঃ \(\frac{1}{5}\sin{\left(5x+\frac{\pi}{3}\right)}+c\)

\(Q.2.(xi)\) \(\int{\sqrt{1-\cos{4x}}dx}\)
উত্তরঃ \(-\frac{1}{\sqrt{2}}\cos{2x}+c\)
[ চঃ২০০৭ ]

\(Q.2.(xii)\) \(\int{\sqrt{1+\cos{x}}dx}\)
উত্তরঃ \(2\sqrt{2}\sin{\frac{x}{2}}+c\)
[ বুয়েটঃ২০০৪-২০০৫ ]

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

\(Q.2.(xiv)\) \(\int{\tan^2{\frac{x}{2}}dx}\)
উত্তরঃ \(2\tan{\frac{x}{2}}-x+c\)

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

\(Q.2.(xvi)\) \(\int{\frac{1-\cos{5x}}{1+\cos{5x}}dx}\)
উত্তরঃ \(\frac{2}{5}\tan{\frac{5x}{2}}-x+c\)
[ সিঃ২০০২ ]

\(Q.2.(xvii)\) \(\int{cosec^2{5x} \ dx}\)
উত্তরঃ \(-\frac{1}{5}\cot{5x}+c\)

\(Q.2.(xviii)\) \(\int{\frac{1}{1+\cos{x}}dx}\)
উত্তরঃ \(\tan{\frac{x}{2}}+c\)
[ কুঃ২০১৩; রাঃ ২০১১, ২০১০; যঃ ২০১১ ]

\(Q.2.(xix)\) \(\int{3\cos{3x}\cos{x}dx}\)
উত্তরঃ \(\frac{3}{8}(\sin{4x}+2\sin{2x})+c\)

\(Q.2.(xx)\) \(\int{\sin{5x}\sin{3x}dx}\)
উত্তরঃ \(\frac{1}{16}(4\sin{2x}-\sin{8x})+c\)
[ যঃ২০১০; বঃ ২০০৮; চঃ২০১২ ]

\(Q.2.(xxi)\) \(\int{\sin{3x}\cos{5x}dx}\)
উত্তরঃ \(\frac{1}{16}(4\cos{2x}-\cos{8x})+c\)
[ সিঃ২০১২; দিঃ২০১২ ]

\(Q.2.(xxii)\) \(\int{7\sin{4x}\sin{2x}dx}\)
উত্তরঃ \(\frac{7}{12}(3\sin{2x}-\sin{6x})+c\)
[ ঢাঃ২০১১; রাঃ২০০৫; যঃ২০০৪ ]

\(Q.2.(xxiii)\) \(\int{5\cos{4x}\sin{3x}dx}\)
উত্তরঃ \(\frac{5}{14}(7\cos{x}-\cos{7x})+c\)
[ সিঃ,দিঃ, রাঃ২০০৯; চঃ২০১১; কুঃ২০১৩; যঃ২০১২,২০১৪ ]
\(Q.2.(xxiv)\) \(\int{4\cos{4x}\sin{5x}dx}\)
উত্তরঃ \(-\frac{2}{9}(\cos{9x}+9\cos{x})+c\)
[ রাঃ২০০৩ ]

\(Q.2.(xxv)\) \(\int{\sin{px}\cos{qx}dx}, (p>q)\)
উত্তরঃ \(-\frac{1}{2}\left\{\frac{\cos{(p+q)x}}{p+q}+\frac{\cos{(p-q)x}}{p-q}\right\}+c\)
[ সিঃ২০০৭; ঢাঃ২০০৩ ]

\(Q.2.(xxvi)\) \(\int{\sin{4x}\sin{2x}dx}\)
উত্তরঃ \(\frac{1}{12}(3\sin{2x}-\sin{6x})+c\)

\(Q.2.(xxvii)\) \(\int{3\sin{3x}\cos{4x}dx}\)
উত্তরঃ \(\frac{3}{14}(7\cos{x}-\cos{7x})+c\)

\(Q.2.(xxviii)\) \(\int{\cos^2{3\theta}d\theta}\)
উত্তরঃ \(\frac{1}{12}(6\theta-\sin{6\theta})+c\)

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

\(Q.2.(xxx)\) \(\int{\cos^2{2x}dx}\)
উত্তরঃ \(\frac{1}{8}(4x+\sin{4x})+c\)
[ ঢঃ২০০০ ]

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

\(Q.2.(xxxii)\) \(\int{\sin^3{\theta}d\theta}\)
উত্তরঃ \(\frac{1}{12}(\cos{3\theta}-9\cos{\theta})+c\)

\(Q.2.(xxxiii)\) \(\int{\cos^3{2x}dx}\)
উত্তরঃ \(\frac{1}{24}(\cos{6x}-9\cos{2x})+c\)
[ ঢাঃ২০০১]

\(Q.2.(xxxiv)\) \(\int{\sin^4{x}dx}\)
উত্তরঃ \(\frac{1}{32}(12x-8\sin{2x}+\sin{4x})+c\)
[ কুঃ২০০৯]

\(Q.2.(xxxv)\) \(\int{\left(1+\cos^2{\frac{x}{2}}\right)dx}\)
উত্তরঃ \(\frac{1}{2}(3x+\sin{x})+c\)
[ বুটেক্সঃ২০০৪-২০০৫ ]

\(Q.2.(xxxvi)\) \(\int{\sin^2{x}\cos{2x}dx}\)
উত্তরঃ \(\frac{1}{16}(4\sin{2x}-\sin{4x}-4x)+c\)
[ রাঃ২০১২,২০১৩; সিঃ২০১১; কুঃ২০০৭,২০১০,২০১৩,২০১৪; চঃ২০০২,২০০৯; যঃ২০০৫ ]

\(Q.2.(xxxvii)\) \(\int{(2\cos{x}+\sin{x})\cos{x}dx}\)
উত্তরঃ \(\frac{1}{4}(4x+2\sin{2x}-\cos{2x})+c\)
[ ঢাঃ২০০৫ ]

\(Q.2.(xxxviii)\) \(\int{\sin^2{x}\cos^2{x}dx}\)
উত্তরঃ \(\frac{1}{32}(4x-\sin{4x})+c\)
[ ঢাঃ২০১৩; বঃ২০০৮; চঃ২০০৬; বুয়েটঃ২০০৩-২০০৪ ]

\(Q.2.(xxxix)\) \(\int{\cos^2{3\theta}d\theta}\)
উত্তরঃ \(\frac{1}{12}(6\theta+\sin{6\theta})+c\)

\(Q.2.(xL)\) \(\int{\cos^3{2x}dx}\)
উত্তরঃ \(\frac{1}{24}(\sin{6x}+9\sin{2x})+c\)

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

\(Q.2.(xLii)\) \(\int{\frac{1}{1+\sin{x}}dx}\)
উত্তরঃ \(\tan{x}-\sec{x}+c\)
[ চঃ২০১০; যঃ২০১৩]

\(Q.2.(xLiii)\) \(\int{\frac{1}{1-\cos{3x}}dx}\)
উত্তরঃ \(-\frac{1}{3}\cot{\frac{3x}{2}}+c\)

\(Q.2.(xLiv)\) \(\int{\frac{1+e^{5x}}{\sqrt{e^{3x}}}dx}\)
উত্তরঃ \(-\frac{2}{3}e^{-\frac{3x}{2}}+\frac{2}{7}e^{\frac{7x}{2}}+c\)

\(Q.2.(xLv)\) \(\int{\frac{1}{1-\sin{x}}dx}\)
উত্তরঃ \(\tan{x}+\sec{x}+c\)
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