Help with $\int_0^\infty x^me^{-ax^n}dx$

Question

I need the solution of the following integral $$\int_0^\infty x^me^{-ax^n}dx$$ where $a,n,m$ are all positive constants with $n\geq 2$. I have searched for it in the Gradshteyn but was unable to find a solution for this.

For $n=2$ I think we can use some thing used for normal distribution but for general values of $n$ I do not know how to solve it.

Answer 1

One may perform the change of variable $$ x=\frac{u^{1/n}}{a^{1/n}},\quad dx=\frac{u^{1/n-1}}{na^{1/n}}du,\quad ax^n=u, $$ obtaining $$ \int_0^\infty x^me^{-ax^n}dx=\frac{a^{\large -\frac{m+1}{n}}}{n}\int_0^\infty u^{(m+1)/n-1} e^{-u}du $$ then, using the standard integral representation of the gamma function, one gets

$$ \int_0^\infty x^me^{-ax^n}dx=\frac{a^{\large -\frac{m+1}{n}} }{n}\Gamma\left(\frac{m+1}{n}\right). $$