If ∫ e^x (xsin^(-1)x/√(1-x^2) + sin^(-1)x/(1-x^2)^(3/2) + x/(1-x^2)) dx = g(x)+C,

Question

If \(\int e^{x}\left(\frac{x \sin ^{-1} x}{\sqrt{1-x^{2}}}+\frac{\sin ^{-1} x}{\left(1-x^{2}\right)^{3 / 2}}+\frac{x}{1-x^{2}}\right) d x=g(x)+C,\) where C is the constant of integration, then \(\mathrm{g}\left(\frac{1}{2}\right)\) equals :

(1) \(\frac{\pi}{6} \sqrt{\frac{e}{2}}\)

(2) \(\frac{\pi}{4} \sqrt{\frac{\mathrm{e}}{2}}\)

(3) \(\frac{\pi}{6} \sqrt{\frac{\mathrm{e}}{3}}\)

(4) \(\frac{\pi}{4} \sqrt{\frac{\mathrm{e}}{3}}\)

Answer 1

Correct option is (3) \(\frac{\pi}{6} \sqrt{\frac{\mathrm{e}}{3}}\) 

\(\because \frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{x} \sin ^{-1} \mathrm{x}}{\sqrt{1-\mathrm{x}^{2}}}\right)=\frac{\sin ^{-1} \mathrm{x}}{\left(1-\mathrm{x}^{2}\right)^{3 / 2}}+\frac{\mathrm{x}}{1-\mathrm{x}^{2}}\)

\(\Rightarrow \int e^{x}\left(\frac{x \sin ^{-1} x}{\sqrt{1-x^{2}}}+\frac{\sin ^{-1} x}{\left(1-x^{2}\right)^{3 / 2}}+\frac{x}{1-x^{2}}\right) d x\)

\(=e^{x} \cdot \frac{x \sin ^{-1} x}{\sqrt{1-x^{2}}}+c=g(x)+C\)

Note : assuming \(g(x)=\frac{x e^{x} \sin ^{-1} x}{\sqrt{1-x^{2}}}\)

\(g(1 / 2)=\frac{\mathrm{e}^{1 / 2}}{2} \cdot \frac{\frac{\pi}{6} \times 2}{\sqrt{3}}=\frac{\pi}{6} \sqrt{\frac{\mathrm{e}}{3}}\)  

Comment : In this question we will not get a unique function \(\mathrm{g}(\mathrm{x}),\) but in order to match the answer we will have to assume g(x)=\(\frac{\mathrm{xe}^{\mathrm{x}} \sin ^{-1} \mathrm{x}}{\sqrt{1-\mathrm{x}^{2}}}.\)