# A general formula for a specific improper integral

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.

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.

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

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

• Is there a specific set of values for m,n and for which it exists or is more of a spectrum? – Marshall Case Mar 5 '15 at 3:52
• Check the existence conditions of the gamma function. – science Mar 5 '15 at 3:54
• I'm a little confused how you used the substitution to arrive at that solution, I used ax^n and ended up with something totally different, that didn't resemble the gamma function as nicely – Marshall Case Mar 5 '15 at 15:14
• @MarshallCase: You should be able to get the answer. Just work the problem step by step. If you still have problems let me know. – science Mar 6 '15 at 3:36