$I=\int\limits_{0}^{\pi }{{{x}^{2}}dx}+\int\limits_{0}^{\pi }{x\sin xdx}$  $=\int\limits_{0}^{\pi }{{{x}^{2}}dx}-\int\limits_{0}^{\pi }{xd\left( \cos x \right)}$$=\frac{{{x}^{3}}}{3}\left| \begin{align}  & \pi  \\  & 0 \\ \end{align} \right.$$-\left( x\cos x \right)\left| \begin{align}  & \pi  \\  & 0 \\ \end{align} \right.$$+\int\limits_{0}^{\pi }{\cos xdx}$

$=\frac{{{\pi }^{3}}}{3}+\pi +\sin x\left| \begin{align}  & \pi  \\ & 0 \\ \end{align} \right.=\frac{1}{3}{{\pi }^{3}}+\pi $