Question

${\int }_0^{1}x{e}^{x}dx=$

Answer 1

$\int x.e^x.dx$

Apply UV formula

$\int u.v=u\int v-\int$$\int v.du$

So $\int \underset{u}{x}.\underset{v}{e^x}.dx=x\int e^x.dx-\int$$(\int e^x.dx).\frac{dx}{dx}.dx$

$=\left(x{e}^x\right)-e^x$

So applying limits

${\int }_0^{1}x.e^x.dx=(x.e^x{)}_0^1-(e^x{)}_0^1$

$=(e^1.1-0)-(e-e^0)$

$=\left(e\right)-(e-1)$

$=1$