অনির্দিষ্ট যোগজীকরণ ( Indefinite integration )

অনুশীলনী \(10.A\) উদাহরণ সমুহ
যোজিত ফল নির্ণয় করঃ
\((1.)\) \(\int{dx}\)
উত্তরঃ \(x+c\)

\((2.)\) \(\int{x(1+\sqrt{x})dx}\)
উত্তরঃ \(\frac{1}{2}x^{2}+\frac{2}{5}x^{\frac{5}{2}}+c\)

\((3.)\) \(\int{\cot^2{\theta}d\theta}\)
উত্তরঃ \(-\cot{\theta}-\theta+c\)

\((4.)\) \(\int{\frac{\cos{\theta}}{\sin^2{\theta}}d\theta}\)
উত্তরঃ \(-cosec \ {\theta}+c\)

\((5.)\) \(\int{\sec^2{x} \ cosec^2{x}dx}\)
উত্তরঃ \(\tan{x}-\cot{x}+c\)

\((6.)\) \(\int{\frac{1}{x^2}dx}\)
উত্তরঃ \(-\frac{1}{x}+c\)

\((7.)\) \(\int{\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)dx}\)
উত্তরঃ \(\frac{2}{3}x^{\frac{3}{2}}-2\sqrt{x}+c\)

\((8.)\) \(\int{\left(1+\frac{1}{x}+\frac{1}{x^2}\right)dx}\)
উত্তরঃ \(x+\ln{|x|}-\frac{1}{x}+c\)

\((9.)\) \(\int{\frac{e^{2x}-e^{4x}}{e^{x}-e^{-x}}dx}\)
উত্তরঃ \(-\frac{1}{3}e^{3x}+c\)

\((10.)\) \(\int{4x^3dx}\)
উত্তরঃ \(x^4+c\)
\((11.)\) \(\int{\left(\sqrt{x}-\frac{1}{\sqrt[3]{x}}\right)dx}\)
উত্তরঃ \(\frac{2}{3}x^{\frac{3}{2}}-\frac{3}{2}x^{\frac{2}{3}}+c\)

\((12.)\) \(\int{\frac{t^5-3t^3+5t}{t^3}dt}\)
উত্তরঃ \(\frac{1}{3}t^3-3t-\frac{5}{t}+c\)

\((13.)\) \(\int{\frac{(x-3)^2}{\sqrt{x}}dx}\)
উত্তরঃ \(\frac{2}{5}x^{\frac{5}{2}}-4x^{\frac{3}{2}}+18\sqrt{x}+c\)

\((14.)\) \(\int{\frac{xe^x}{(x+1)^2}dx}\)
উত্তরঃ \(\frac{e^x}{x+1}+c\)

\((15.)\) \(\int{\frac{at^2+bt+c}{t}dt}\)
উত্তরঃ \(\frac{1}{2}at^2+bt+c\ln{|t|}+k\)

\((16.)\) \(\int{\frac{1-\cos{2\theta}}{1+\cos{2\theta}}d\theta}\)
উত্তরঃ \(\tan{\theta}-\theta+c\)
[ যঃ ২০১৪; বঃ২০০৩ ]

\((17.)\) \(\int{(ax^3+bx^2+cx)dx}\)
উত্তরঃ \(\frac{1}{4}ax^4+\frac{1}{3}bx^3+\frac{1}{2}cx^2+k\)

\((18.)\) \(\int{(3\cos{x}-5\sec^2{x})dx}\)
উত্তরঃ \(3\sin{x}-5\tan{x}+c\)

\((19.)\) \(\int{\frac{t^2+3t+1}{\sqrt{t}}dt}\)
উত্তরঃ \(\frac{2}{5}t^{\frac{5}{2}}+2t^{\frac{3}{2}}+2\sqrt{t}+c\)
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