Mathematics - Indefinite Integration Question with Solution | TestHub

$\mathrm{I}=\int \frac{{\mathrm{x}}^2\left(1-\frac{1}{{\mathrm{x}}^2}\right)\mathrm{dx}}{{\mathrm{x}}^2\left(\mathrm{x}+\frac{1}{\mathrm{x}}\right){\left({\mathrm{x}}^2+\frac{1}{{\mathrm{x}}^2}\right)}^{1/2}}$ Let $\mathrm{x}+\frac{1}{\mathrm{x}}=\mathrm{p}\Rightarrow \left(1-\frac{1}{{\mathrm{x}}^2}\right)\mathrm{dx}=\mathrm{dp}$ $\mathrm{I}=\int \frac{\mathrm{dp}}{\mathrm{p}\sqrt{{\mathrm{p}}^2-2}}=\frac{1}{\sqrt{2}}{\sec }^{-1}\frac{\mathrm{p}}{\sqrt{2}}$ $=\frac{1}{\sqrt{2}}{\sec }^{-1}\left(\frac{{\mathrm{x}}^2+1}{\sqrt{2}\mathrm{x}}\right)+\mathrm{c}$