Integrate : $\int e^x \bigg( \sin ^{-1} x +\large\frac{1}{\sqrt {1-x^2}}\bigg) dx $

Question

$(a)\;e^x \sin ^{-1} x +c \\(b)\;\sin ^{-1}+c \\(c)\;x+c \\ (d)\;None$

Answer 1

$\int e^x \bigg( \sin ^{-1} x +\large\frac{1}{\sqrt {1-x^2}}\bigg) dx $

=> $ \int e^x . \sin ^{-1} x dx + e^x. \large\frac{1}{\sqrt {1-x^2}}$$dx$

=> $\sin ^{-1}x.e^x -\int \large\frac{1}{\sqrt {1+x^2}}$$ \times e^x dx \int e^x \times \large\frac{1}{\sqrt {1-x^2}}$$dx$

$e^x \sin ^{-1} x +c$

Hence a is the correct answer.