$I=\int\ x\cdot \sin x \mathrm dx+\int \cos^4x\cdot\sin x \mathrm dx$$=-\int x \mathrm d\left( \cos x\right)-\int\cos^4x\mathrm d\left(\cos x\right)$
$=-x\cdot \cos x+\int\cos x\mathrm dx-\frac{\cos^5x}{5}$
$=-x\cdot\cos x+\sin x-\frac{\cos^5x}{5}+C$