A general formula for a specific improper integral

Question

The integral I'm after is here:

The question is a little ambiguous whether it wants a general solution for this, but based on thought, I would guess there are many different answers based on the relationship between x, n and a. It also asks for the specific case where m=5, a=4, and n=4. Thanks in advance.

Answer 1

The integral admits the closed form

$$ I = \frac{a^{-\frac{1+n}{n}}}{n}\Gamma\left( \frac{m+1}{n} \right), $$

which can be evaluated using the change of variables $ax^n=t$. $\Gamma(z)$ is the gamma function. Pay attention for what $m$ and $n$ the integral exists.

Note: The gamma function is defined as

$$ \Gamma(z) = \int_{0}^{\infty}t^{z-1}e^{-t}dt,\quad Re(z)>0. $$

Answer 2

Doing the same as science proposed (which is indeed the simplest way), I instead arrive to $$I=\frac{a^{-\frac{m+1}{n}} }{n}\Gamma \left(\frac{m+1}{n}\right)$$ provided some conditions I let you finding.