Integral Mixed with trigonometry

Exercise:

Integrate, $\int{\cos x{{\sin }^{2}}x{{e}^{\sin x}}dx}$

Solution: Let $u=\sin x\Rightarrow du=\cos x\,dx$ , so $\int{\cos x{{\sin }^{2}}x{{e}^{\sin x}}dx}=\int{{{u}^{2}}{{e}^{u}}du}$

Now apply by parts integrations let $U={{u}^{2}}\,\And \,dW={{e}^{u}}du$ $\Rightarrow dU=2u\,du\,\And \,W={{e}^{u}}$

$\int{{{u}^{2}}{{e}^{u}}du}={{u}^{2}}{{e}^{u}}-2\int{u{{e}^{u}}du}$ Again take $y=u\,\And \,df={{e}^{u}}du$ $\Rightarrow dy=du\,\And \,f={{e}^{u}}$

$={{u}^{2}}{{e}^{u}}-2\left( u{{e}^{u}}-\int{{{e}^{u}}du} \right)={{u}^{2}}{{e}^{u}}-2u{{e}^{u}}+2{{e}^{u}}+c={{e}^{\sin x}}\left( {{\sin }^{2}}x-2\sin x+2 \right)+c$