Question

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

Answer 1

${\int }_0^{1}x{e}^{x}dx=1$
Using integration by parts
$={[x\int e^x-\int \left(\frac{d}{dx}\right(x)\int e^{x}dx)dx]}_0^1$
$={[x{e}^x-\int (1.e^x\left)dx\right]}_0^1$
$=[x{e}^x-e^x{]}_0^1$
$=\left[\right(1e^1-e^1)-(0-e^0\left)\right]$
$=1$
Hence proved.