Question

The value of $(\int x.e^{-x}dx)$ is,

Answer 1

$\int x.e^{-x}dx$

$\Rightarrow$ Using $uv$ rule

consider (LATE) $u=x,v=e^{-x}$

$\int uvdx$

$=u\int vdx-\int u^3(\int vdx)dx$

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

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

$=-x^{e^{-x}}-e^{-x}$

$=e^{-x}(-x-1)$