Question

integration of xsin^​-1 x² dx / root(1 - x⁴)

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{I}=\int \frac{\mathrm{x}{\sin }^{-1}\left({\mathrm{x}}^2\right)}{\sqrt{1-{\mathrm{x}}^4}}\mathrm{dx} \\ \mathrm{Put}{\sin }^{-1}\left({\mathrm{x}}^2\right)=\mathrm{t} \\ \Rightarrow \frac{2\mathrm{x}}{\sqrt{1-{\mathrm{x}}^4}}\mathrm{dx}=\mathrm{dt} \\ \Rightarrow \frac{\mathrm{xdx}}{\sqrt{1-{\mathrm{x}}^4}}=\frac{\mathrm{dt}}{2} \\ \mathrm{Now},\mathrm{I}=\frac{1}{2}\int \mathrm{t}\mathrm{dt} \\ =\frac{1}{2}\times \frac{{\mathrm{t}}^2}{2}+\mathrm{C} \\ =\frac{{\mathrm{t}}^2}{4}+\mathrm{C} \\ =\frac{{\left[{\sin }^{-1}\left({\mathrm{x}}^2\right)\right]}^2}{4}+\mathrm{C} \end{gathered}$$