JEE Math Practice Question
If $\int \mathrm{e}^x\left(\frac{x \sin ^{-1} x}{\sqrt{1-x^2}}+\frac{\sin ^{-1} x}{\left(1-x^2\right)^{3 / 2}}+\frac{x}{1-x^2}\right) \mathrm{d} x=\mathrm{g}(x)+\mathrm{C}$, where C is the constant of integration, then $g\left(\frac{1}{2}\right)$ equals :
- A.$\frac{\pi}{6} \sqrt{\frac{\mathrm{e}}{3}}$
- B.$\frac{\pi}{6} \sqrt{\frac{\mathrm{e}}{2}}$
- C.$\frac{\pi}{4} \sqrt{\frac{\mathrm{e}}{3}}$
- D.$\frac{\pi}{4} \sqrt{\frac{\mathrm{e}}{2}}$
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