Question:
Evaluate the following integrals:
$\int e^{x} \cdot \frac{\sqrt{1-x^{2}} \sin ^{-1} x+1}{\sqrt{1-x^{2}}} d x$
Solution:
Let $I=\int e^{x} \frac{\sqrt{1-x^{2}} \sin ^{-1} x+1}{\sqrt{1-x^{2}}} d x$
$I=\int e^{x} \sin ^{-1} x+\int e^{x} \frac{1}{\sqrt{1-x^{2}}} d x$
Integrating by parts
$=e^{x} \sin ^{-1} x-\int e^{x}\left(\frac{d}{d x}\left(\sin ^{-1} x\right)\right) d x+\int e^{x} \frac{1}{\sqrt{1-x^{2}}} d x$
$=e^{x} \sin ^{-1} x-\int e^{x} \frac{1}{\sqrt{1-x^{2}}} d x+\int e^{x} \frac{1}{\sqrt{1-x^{2}}} d x$
$=e^{x} \sin ^{-1} x+c$
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