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Q. Integrate $ \frac{{{\sec }^{2}}\,({{\sin }^{-1}}x)}{\sqrt{1-{{x}^{2}}}} $

J & K CETJ & K CET 2014Integrals

Solution:

Let $ l=\int{\frac{{{\sec }^{2}}({{\sin }^{-1}}x)}{\sqrt{1-{{x}^{2}}}}}dx $ Again, let $ {{\sin }^{-1}}x=t $
$ \Rightarrow $ $ \frac{dt}{dx}=\frac{1}{\sqrt{1-{{x}^{2}}}} $
$ \Rightarrow $ $ dt=\frac{1}{\sqrt{1-{{x}^{2}}}}\,dx $
$ \therefore $ $ l=\int{{{\sec }^{2}}\,t\,dt} $
$=\tan \,t+C $
$=\tan ({{\sin }^{-1}}x)+C $