Mathematics - Indefinite Integration Question with Solution | TestHub

$\begin{array}{rl}& \frac{d}{dx}\left(\frac{x\cdot {\sin }^{-1}x}{\sqrt{1-x^2}}\right)-{\sin }^{-1}x\cdot \left(\frac{1\cdot \sqrt{1-x^2}+\frac{x\cdot 2x}{2\sqrt{1-x^2}}}{1-x^2}\right)\\ & =\frac{x}{\sqrt{1-x^2}}\cdot \frac{1}{\sqrt{1-x^2}}\\ & =\frac{{\sin }^{-1}x}{{\left(1-x^2\right)}^{3/2}}+\frac{x}{1-x^2}\end{array}$ $\begin{array}{rl}& \text{ Hence, }I=\int e^x\left(f(x)+f^{\prime }(x)\right)dx\\ & =e^x\cdot f(x)+C\\ & I=e^x\cdot \frac{x\cdot {\sin }^{-1}x}{\sqrt{1-x^2}}+C=g(x)+C\\ & \Rightarrow g(x)=\frac{x{e}^x{\sin }^{-1}x}{\sqrt{1-x^2}}\text{ and }g(1/2)=\frac{\pi }{6}\sqrt{\frac{e}{3}}\end{array}$