QJEEM25Ma6a5a: JEE Main 2025 Mathematics - Class 12 JEEM25Ma6a5a Solution: If , where C is the constant of integration, then equals :

$\begin{aligned}&\begin{aligned}& \because \frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{x} \sin ^{-1} \mathrm{x}}{\sqrt{1-\mathrm{x}^2}}\right)=\frac{\sin ^{-1} \mathrm{x}}{\left(1-\mathrm{x}^2\right)^{3 / 2}}+\frac{\mathrm{x}}{1-\mathrm{x}^2} \\& \Rightarrow \int \mathrm{e}^{\mathrm{x}}\left(\frac{\mathrm{x} \sin ^{-1} \mathrm{x}}{\sqrt{1-\mathrm{x}^2}}+\frac{\sin ^{-1} \mathrm{x}}{\left(1-\mathrm{x}^2\right)^{3 / 2}}+\frac{\mathrm{x}}{1-\mathrm{x}^2}\right) \mathrm{dx} \\& =\mathrm{e}^{\mathrm{x}} \cdot \frac{\mathrm{x} \sin ^{-1} \mathrm{x}}{\sqrt{1-\mathrm{x}^2}}+\mathrm{c}=\mathrm{g}(\mathrm{x})+\mathrm{C}\end{aligned}\\&\text { Note : assuming } g(x)=\frac{\mathrm{xe}^{\mathrm{x}} \sin ^{-1} x}{\sqrt{1-\mathrm{x}^2}}\\&g(1 / 2)=\frac{\mathrm{e}^{1 / 2}}{2} \cdot \frac{\frac{\pi}{6} \times 2}{\sqrt{3}}=\frac{\pi}{6} \sqrt{\frac{\mathrm{e}}{3}}\end{aligned}$

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 $g(x)=\frac{\mathrm{xe}^{\mathrm{x}} \sin ^{-1} \mathrm{x}}{\sqrt{1-\mathrm{x}^2}}$ .