Question

$\underset{0}{\overset{\sqrt{2}}{\int }}{x}^3{e}^{x^2}dx$

Answer 1

$\int x^3{e}^{x^2}dx$

Let $x^2=u$

Then $2xdx=du$

When $x=0,u=0$ and when $x=\sqrt{2},u=2$

$\int x^3{e}^{x^2}dx$

$=\int x^2{e}^{x^2}xdx=\frac{1}{2}\int u{e}^{u}du$

Using integration by parts,

$\frac{1}{2}\int u{e}^{u}du=\frac{1}{2}(u{e}^u-\int 1.e^{u}du)$

$=\frac{1}{2}(u{e}^u-e^u)$

Applying limits,

${\int }_0^{2}u{e}^{u}du=\frac{1}{2}[2e^2-e^2+1]=\frac{1}{2}[e^2+1]$