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Questions  >>  JEEMAIN and NEET  >>  Mathematics  >>  Class12  >>  Integral Calculus
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Q)

Integrate : $\int \large \frac{e^{\Large\cos ^{-1}x}}{\sqrt {1-x^2}}$$dx$

$(a)\;-e^{\pi/2}-\sin^{-1}x \\(b)\;-e^{\pi/2}-\sin ^{-1} x \\(c)\;-e^{\pi/2}- \cos ^{-1}x \\ (d)\;None$

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A)
$\cot^{-1} x=\large\frac{\pi}{2}$$ -\sin ^{-1} x$
$\int \large\frac{e^{\pi/2-\sin^{-1}x }}{\sqrt {1-x^2}}$
=> $\sin ^{-1} x =t$
=>$\large\frac{1}{\sqrt {1-x^2}} $$dx=dt$
=> $ \int e^{\pi/2-t}.dt$
=> $e^{\pi/2-t}(-1) +c$
$-e^{\pi/2}-\sin ^{-1} x$
Hence b is the correct answer.
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