Question

Solve:
$\int x^2{e}^x\sin xdx$

Answer 1

$I=\int x^2{e}^x\sin xdx$

Integrating by parts $u=x^2.V=e^x\sin x$

$u^1=2x,\int vdx=\frac{1}{2}{e}^x(\sin x-\cos x)$ [Integrated by parts]

$I=x^2\int e^x\sin dx-\int 2x(\int e^x\sin xdx)dx$

$=\frac{x^2{e}^x}{2}(\sin x-\cos x)-\int x{e}^x(\sin x-\cos x)dx$

Now, $\int e^x(\cos x-\sin x)dx=e^x\cos x[\because \int e^{x}f(x)+f^{\prime }(x)dx=e^{x}f(x\left)\right]$

$I=\frac{x^2{e}^x}{2}(\sin x-\cos x)+[x{e}^x\cos x-\int (1\left)e^x\cos xdx\right]$ [Using by parts]

$=\frac{x^2{e}^x}{2}(\sin x-\cos x)+x{e}^x\cos x-\frac{e^x}{2}(\sin x+\cos x)+C$ [Using by parts]

$=\frac{e^x}{2}\left[\sin \right(x\left)\right(x^2-1)+\cos x(-x^2+2x-1\left)\right]+C$

$I=\frac{e^x}{2}\left(\right(x^2-1)\sin x-(x-1{)}^2\cos x)+C$