Integrate the function $x\cos\left(x\right)$ from 0 to $\frac{\pi }{2}$

Evaluate the integral

$\int_0^{\infty}\left(xe^{-x}\right)dx$

Evaluate the integral

$\int_{-\infty}^0\left(xe^x\right)dx$

Evaluate the integral

$\int_0^{\infty}\left(1-x\right)e^{-x}dx$

Evaluate the integral

$\int_{-\infty}^0xe^xdx$

Evaluate the integral

$\int_0^1\left(x\cdot\ln\left(x+1\right)\right)dx$

Evaluate the integral

$\int_0^1xln\left(x\right)dx$

Evaluate the integral

$\int_0^1\left(\ln\left(1+x\right)\right)dx$