defence exam Agniveer Vayu Group X & XY Mock Test 01/2026 Mathematics Integral Calculus Indefinite Integrals
If \(\rm \int e^{x}\left(\frac{x \sin ^{-1} x}{\sqrt{1-x^{2}}}+\frac{\sin ^{-1} x}{\left(1-x^{2}\right)^{3 / 2}}+\frac{x}{1-x^{2}}\right) d x=g(x)+C\), where C is the constant of integration, then g\(\left(\frac{1}{2}\right)\) equals :
1
\(\frac{\pi}{6} \sqrt{\frac{\mathrm{e}}{2}}\)
2
\(\frac{\pi}{4} \sqrt{\frac{\mathrm{e}}{2}}\)
3
\(\frac{\pi}{6} \sqrt{\frac{\mathrm{e}}{3}}\)
4
\(\frac{\pi}{4} \sqrt{\frac{\mathrm{e}}{3}}\)