Question

what is the integration of V/(1-V²) ? explain briefly.

Answer 1

$$\begin{gathered} \mathrm{I}=\int \frac{\mathrm{V}}{1-{\mathrm{V}}^2}\mathrm{dV}.....\left(1\right) \\ \mathrm{put}1-{\mathrm{V}}^2=\mathrm{t} \\ \Rightarrow -2\mathrm{VdV}=\mathrm{dt}\Rightarrow \mathrm{VdV}=-\frac{1}{2}\mathrm{dt} \\ \mathrm{Substitute}\mathrm{these}\mathrm{values}\mathrm{in}\mathrm{equation}\left(1\right), \\ \mathrm{I}=\int -\frac{1}{2\mathrm{t}}\mathrm{dt} \\ =-\frac{1}{2}\mathrm{logt}+\mathrm{C} \\ =-\frac{1}{2}\log \left(1-{\mathrm{V}}^2\right)+\mathrm{C} \\ \Rightarrow \mathbf{I}\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{log}\left(\mathbf{1}\mathbf{-}{\mathbf{V}}^{\mathbf{2}}\right)\mathbf{+}\mathbf{C} \end{gathered}$$