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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class12  >>  Integral Calculus
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Integrate : $\int \large\frac{\sin ^{-1} x }{(1-x^2)^{3/2}}$$dx$

$(a)\;(\sin ^{-1}x) (1-x^2)^{3/2}+c \\(b)\;\sin ^{-1}x \tan (\sin ^{-1}x)+\log (\cos (\sin ^{-1}x)+c \\(c)\;\sin ^{-1}x-\cos ^{-1}x+c \\ (d)\;None$
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1 Answer

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$\sin ^{-1} x =t => \large\frac{1}{\sqrt {1-x^2}}dx=dt$
$\int \large\frac{t}{(1-\sin ^2 t)}.$$dt$
$I= \int t. \sec^2 t.dt$
After solve it
$I= t \tan t -\int \tan t$
$\quad= t.\tan t +\log |\cos t|+c$
$\quad=\sin ^{-1}x \tan (\sin ^{-1}x)+\log (\cos (\sin ^{-1}x)+c$
Hence b is the correct answer.
answered Dec 30, 2013 by meena.p
 
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