A) \[\sin \,({{\tan }^{-1}}x)+C\]
B) \[\tan \,({{\sec }^{-1}}x)+C\]
C) \[\tan \,({{\sin }^{-1}}x)+C\]
D) \[-\tan \,(co{{s}^{-1}}x)+C\]
Correct Answer: C
Solution :
Let \[l=\int{\frac{{{\sec }^{2}}({{\sin }^{-1}}x)}{\sqrt{1-{{x}^{2}}}}}dx\] Again, let \[{{\sin }^{-1}}x=t\] \[\Rightarrow \] \[\frac{dt}{dx}=\frac{1}{\sqrt{1-{{x}^{2}}}}\] \[\Rightarrow \] \[dt=\frac{1}{\sqrt{1-{{x}^{2}}}}\,dx\] \[\therefore \] \[l=\int{{{\sec }^{2}}\,t\,dt}\] \[=\tan \,t+C\] \[=\tan ({{\sin }^{-1}}x)+C\]
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