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

অনুশীলনী \(10.C\) উদাহরণ সমুহ
যোজিত ফল নির্ণয় করঃ
\((1.)\) \(\int{x^2(3-2x^3)^4dx}\)
উত্তরঃ \(-\frac{1}{30}(3-2x^3)^{5}+c\)

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

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

\((4.)\) \(\int{cosec \ {x}\cot{x}e^{cosec \ {x}}dx}\)
উত্তরঃ \(-e^{cosec \ {x}}+c\)

\((5.)\) \(\int{\frac{\sec{x}}{\ln{|\sec{x}+\tan{x}|}}dx}\)
উত্তরঃ \(\ln{(\ln{|\sec{x}+\tan{x}|})}+c\)

\((6.)\) \(\int{\frac{cosec \ {x}}{\ln{|cosec \ {x}-\cot{x}|}}dx}\)
উত্তরঃ \(\ln{(\ln{|cosec \ {x}-\cot{x}|})}+c\)

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

\((8.)\) \(\int{\tan^3{x}dx}\)
উত্তরঃ \(\frac{1}{2}\tan^2{x}-\ln{|\sec{x}|}+c\)

\((9.)\) \(\int{\frac{\sin{2x}}{a+b\sin^2{x}}dx}\)
উত্তরঃ \(\frac{1}{b}\ln{|a+b\sin^2{x}|}+c\)
\((10.)\) \(\int{\frac{\sin{(\tan^{-1}{x})}}{1+x^2}dx}\)
উত্তরঃ \(-\frac{1}{\sqrt{1+x^2}}+c\)

\((11.)\) \(\int{\frac{\cos{(\tan^{-1}{x})}}{1+x^2}dx}\)
উত্তরঃ \(\frac{x}{\sqrt{1+x^2}}+c\)

\((12.)\) \(\int{\frac{\sec{(\cot^{-1}{x})}}{1+x^2}dx}\)
উত্তরঃ \(-\ln{\left|\frac{x^2+2}{x}\right|}+c\)

\((13.)\) \(\int{\frac{2x\tan^{-1}{x^2}}{1+x^4}dx}\)
উত্তরঃ \(\frac{1}{2}\left(\tan^{-1}{x^2}\right)^2+c\)

\((14.)\) \(\int{\frac{\tan^{-1}{\sqrt{x}}}{\sqrt{x}(1+x)}dx}\)
উত্তরঃ \((\tan^{-1}{\sqrt{x}})^2+c\)

\((15.)\) \(\int{\frac{\tan^{-1}{\sqrt[3]{x}}}{\sqrt[3]{x^2}(1+\sqrt[3]{x^2})}dx}\)
উত্তরঃ \(\frac{3}{2}(\tan^{-1}{\sqrt[3]{x}})^2+c\)

\((16.)\) \(\int{\frac{\sec^{-1}{(\sqrt{x})}}{x\sqrt{x-1}}dx}\)
উত্তরঃ \(\{\sec^{-1}{(\sqrt{x})}\}^2+c\)

\((17.)\) \(\int{\frac{cosec^{-1}{(\sqrt{x})}}{x\sqrt{x-1}}dx}\)
উত্তরঃ \(-\{cosec^{-1}{(\sqrt{x})}\}^2+c\)
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