_{${\displaystyle \underset{0}{\overset{\pi }{\int }}\frac{{\sin }^{3}x}{1+{\cos }^{2}x}dx=}$}

_{$I={\displaystyle \underset{0}{\overset{\pi }{\int }}\left(\frac{1-Co{s}^{2}x}{1+Co{s}^{2}x}\right)}Sinxdx$      $\begin{array}{c}\cos x=t,\\ -\sin xdx=dt\end{array}$}

    _{$={\displaystyle \underset{1}{\overset{-1}{\int }}\left(\frac{1-t^2}{1+t^2}\right)}\left(-dt\right)$}

    _{$={\displaystyle \underset{-1}{\overset{1}{\int }}\frac{1-t^2}{1+t^2}dt}$}

    _{$=2{\displaystyle \underset{0}{\overset{1}{\int }}\frac{1-t^2}{1+t^2}dt}$}

     _{$=2{\displaystyle \underset{0}{\overset{1}{\int }}\frac{2-\left(1+t^2\right)}{1+t^2}}$}

     _{$=2{\left(2Ta{n}^{-1}t-t\right)}_0^1$}

     _{$=2\left(2.\frac{\pi }{4}-1\right)$}

    _{$=\pi -2$}