অনির্দিষ্ট যোগজীকরণ ( Indefinite integration )

অনুশীলনী \(10.A / Q.2\)-এর প্রশ্নসমুহ

যোজিত ফল নির্ণয় করঃ
\(Q.2.(i)\) \(\int{\tan^2{x}dx}\)
উত্তরঃ \(\tan{x}-x+c\)
[ ঢাঃ ২০০৫; কুঃ২০০৭ ]

\(Q.2.(ii)\) \(\int{\frac{1-\cos{2x}}{1+\cos{2x}}dx}\)
উত্তরঃ \(\tan{x}-x+c\)
[ যঃ ২০১৪; বঃ২০০৩ ]

\(Q.2.(iii)\) \(\int{\frac{\sec{x}-\cos{x}}{\cos{x}}dx}\)
উত্তরঃ \(\tan{x}-x+c\)

\(Q.2.(iv)\) \(\int{\cot^2{x}dx}\)
উত্তরঃ \(-\cot{x}-x+c\)

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

\(Q.2.(vi)\) \(\int{\sec{x}(\sec{x}-\tan{x})dx}\)
উত্তরঃ \(\tan{x}-\sec{x}+c\)
[ বঃ ২০১৩ ]

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

\(Q.2.(viii)\) \(\int{\sqrt{1-\cos{2x}}dx}\)
উত্তরঃ \(-\sqrt{2}\cos{x}+c\)
[ চঃ২০১২,২০০৯,২০০৫; বঃ২০০৮; সিঃ ২০০৬ ]

\(Q.2.(ix)\) \(\int{\frac{dx}{1-\sin{x}}}\)
উত্তরঃ \(\tan{x}+\sec{x}+c\)
[ ঢাঃ২০০৭; যঃ ২০১৫; সিঃ২০০৩ ]

\(Q.2.(x)\) \(\int{\frac{dx}{1+\sin{x}}}\)
উত্তরঃ \(\tan{x}-\sec{x}+c\)
[ রুয়েটঃ২০০৮-২০০৯; বুয়েটঃ ২০০৩-২০০৪; যঃ২০১৩ ]

\(Q.2.(xi)\) \(\int{\frac{1}{1+\cos{2x}}dx}\)
উত্তরঃ \(\frac{1}{2}\tan{x}+c\)
[ কুঃ২০০৮ ]

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

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

\(Q.2.(xiv)\) \(\int{\frac{a\sin^3{x}+b\cos^3{x}}{\sin^2{x}\cos^2{x}}dx}\)
উত্তরঃ \(a\sec{x}-b \ cosec \ {x}+c\)

\(Q.2.(xv)\) \(\int{\left(\sin{\frac{x}{2}}+\cos{\frac{x}{2}}\right)^2dx}\)
উত্তরঃ \(x-\cos{x}+c\)
[ রাঃ২০১৭ ]

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

\(Q.2.(xvii)\) \(\int{(2\cos{x}+\sin{x})\cos{x}dx}\)
উত্তরঃ \(x+\frac{1}{2}\sin{2x}-\frac{1}{4}\cos{2x}+c\)
[ ঢাঃ ২০০৫ ]

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

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

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

\(Q.2.(xxi)\) \(\int{\frac{\cos{\theta}}{\sin^2{\theta}}d\theta}\)
উত্তরঃ \(-cosec \ {\theta}+c\)

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

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

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

\(Q.2.(xxv)\) \(\int{\frac{1}{\sin^2{x}\cos^2{x}}dx}\)
উত্তরঃ \(-2\cot{2x}+c\)
[ ঢাঃ২০১৩; রাঃ২০১৩; যঃ ২০০৮ ]

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

\(Q.2.(xxvii)\) \(\int{\sin{px}\cos{qx}dx}\)
উত্তরঃ \(-\frac{1}{2}\left\{\frac{\cos{(p+q)x}}{p+q}+\frac{\cos{(p-q)x}}{p-q}\right\}+c\)
[ চঃ২০০৮ ]

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

\(Q.2.(xxix)\) \(\int{\sin{4x}\sin{2x}dx}\)
উত্তরঃ \(\frac{1}{12}(3\sin{2x}-\sin{6x})+c\)
[ ঢঃ২০১১ ]

\(Q.2.(xxx)\) \(\int{\sin{5x}\sin{3x}dx}\)
উত্তরঃ \(\frac{1}{16}(4\sin{2x}-\sin{8x})+c\)
[ চঃ২০১২; যঃ২০১০; বঃ২০১২ ]
\(Q.2.(xxxi)\) \(\int{5\cos{4x}\sin{3x}dx}\)
উত্তরঃ \(\frac{5}{14}(7\cos{x}-\cos{7x})+c\)
[ চঃ২০১৫, ২০১১; কুঃ২০১৩; বঃ২০১৪,২০১০; সিঃ২০১৪; দিঃ২০১৪ ]

\(Q.2.(xxxii)\) \(\int{\cos{4x}\sin{4x}dx}\)
উত্তরঃ \(-\frac{1}{16}\cos{8x}+c\)
[ যঃ২০১৪ ]

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

\(Q.2.(xxxiv)\) \(\int{\sin{x^{o}}dx}\)
উত্তরঃ \(-\frac{180}{\pi}\cos{\left(\frac{\pi{x}}{180}\right)}+c\)

\(Q.2.(xxxv)\) \(\int{10^{3x}dx}\)
উত্তরঃ \(\frac{10^{3x}}{3\ln{10}}+c\)

\(Q.2.(xxxvi)\) \(\int{\frac{\tan{x}}{\sqrt{2}\cot{x}}dx}\)
উত্তরঃ \(\frac{1}{\sqrt{2}}(\tan{x}-x)+c\)

\(Q.2.(xxxvii)\) \(\int{\frac{\cos{2x}}{\sin^2{x}\cos^2{x}}dx}\)
উত্তরঃ \(-2cosec \ {2x}+c\)

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

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

\(Q.2.(xL)\) \(\int{\sin^3{x}dx}\)
উত্তরঃ \(\frac{1}{12}(\cos{3x}-9\cos{x})+c\)

\(Q.2.(xLi)\) \(\int{\cos^4{x}dx}\)
উত্তরঃ \(\frac{1}{32}(12x+8\sin{2x}+\sin{4x})+c\)
[ ঢাঃ২০১৪; রাঃ ২০১৪ ]

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

\(Q.2.(xLiii)\) \(\int{\sin^2{x}\cos{2x}dx}\)
উত্তরঃ \(\frac{1}{16}(4\sin{2x}-4x-\sin{4x})+c\)
[ কুঃ২০১৪; সিঃ২০১১ ]

\(Q.2.(xLiv)\) দেখাও যে, \(\int{\sin{x}\cos{x}dx}=-\frac{1}{4}\cos{2x}+c\) বা \(\frac{1}{2}\sin^2{x}+c\) বা \(-\frac{1}{2}\cos^2{x}+c\)
[ ঢাবিঃ২০০৮-২০০৯ ]

\(Q.2.(xLv)\) \(\int{(x^3-5e^x+3\sin{x})dx}\)
উত্তরঃ \(\frac{1}{4}x^4-5e^x-3\cos{x}+c\)

\(Q.2.(xLvi)\) \(\int{(3\cos{x}-5\sec^2{x}+7)dx}\)
উত্তরঃ \(3\sin{x}-5\tan{x}+7x+c\)

\(Q.2.(xLvii)\) \(\int{(\sec{x}\tan{x}-3cosec^2{x})dx}\)
উত্তরঃ \(\sec{x}+3\cot{x}+c\)

\(Q.2.(xLviii)\) \(\int{\frac{\sin{x}+cosec \ {x}}{5\tan{x}}dx}\)
উত্তরঃ \(\frac{1}{5}(\sin{x}-cosec \ {x})+c\)

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

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

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

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

\(Q.2.(Liii)\) \(\int{3\sin{3x}\cos{4x}dx}\)
উত্তরঃ \(\frac{3}{14}(7\cos{x}-\cos{7x})+c\)
[ বঃ২০১০ ]

\(Q.2.(Liv)\) \(\int{4\cos{4x}\sin{5x}dx}\)
উত্তরঃ \(-\frac{2}{9}(\cos{9x}+9\cos{x})+c\)
[ রাঃ২০০৩ ]

\(Q.2.(Lv)\) \(\int{5\cos{5x}\sin{4x}dx}\)
উত্তরঃ \(\frac{5}{18}(9\cos{x}-\cos{9x})+c\)
[ ঢাঃ২০০৬; দিঃ,সিঃ২০১৪ ]

\(Q.2.(Lvi)\) \(Q.2.(Lvi)\) \(\int{\sin{5x}dx}\) এর যোজিত ফল নির্ণয় কর।
উত্তরঃ \(-\frac{1}{5}\cos{5x}+c\)
[ সিঃ২০০৫ ]

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

\(Q.2.(Lviii)\) \(\int{(4\sin{x}+3\cos{x})dx}\)
উত্তরঃ \(3\sin{x}-4\cos{x}+c\)

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

\(Q.2.(Lx)\) \(\int{\frac{\cos{\theta}-\cos{2\theta}}{1-\cos{\theta}}d\theta}\)
উত্তরঃ \(\theta+2\sin{\theta}+c\)
1 2 3 4 5

Leave a Reply