$\begin{align}  & \int{x\sqrt{{{a}^{2}}-{{x}^{2}}}}dx \\ & \text{Let }u={{a}^{2}}-{{x}^{2}} \\ & \frac{du}{dx}=-2x,\text{ }dx=-\frac{du}{2x} \\ & \int{x\sqrt{{{a}^{2}}-{{x}^{2}}}dx}=\int{x({{u}^{\tfrac{1}{2}}})(-\tfrac{du}{2x})}=-\frac{1}{2}\int{{{u}^{\tfrac{1}{2}}}du} \\ & \int{x\sqrt{{{a}^{2}}-{{x}^{2}}}dx}=-\frac{1}{2}\left[ \frac{{{u}^{\tfrac{3}{2}}}}{\tfrac{3}{2}} \right]+C=-\frac{1}{3}{{u}^{\tfrac{3}{2}}}+C \\ & \int{x\sqrt{{{a}^{2}}-{{x}^{2}}}dx}=-\frac{1}{3}\sqrt{{{({{a}^{2}}-{{x}^{2}})}^{3}}}+C \\\end{align}$