Compute the integral $ \int_0^\infty e^{-u^\ell} u^m du $

Question

I have a general formula, $$ \int_0^\infty e^{-u^\ell} u^m du $$ that I wish to solve. I have seen that that the solution is some combination of incomplete Gamma functions in most cases, but I would appreciate a little bit of in-depth explanation as to how to get there.

To clarify, $u, \ell, m$ are all real.

Thanks!

EDIT: In addition to this, $u$ is always positive.

Answer 1

Let $u^l=x, u=x^{1/l}, du=\frac1lx^{1/l-1}dx$

$$ \int_0^\infty e^{-u^l} u^m du=\frac1l\int_0^\infty e^{-x} x^{\frac{m+1}l-1} ~dx=\frac1l\cdot\Gamma\left(\frac{m+1}l\right) $$