# Integral with several parameters

Let $r>0$. Find $(p,q) \in \mathbb{R}^{2}$ such that the integral: $$\int_{1}^{\infty}{\frac{(x^{r}-1)^{p}}{x^{q}}} ~dx$$ converges and for those values calculate it.

I've already calculated the values for which it is convergent. The integral above converges iff $rp-q <-1$ or equivalently when $p<\frac{q-1}{r}$.

I get stuck when trying to calculate it. I've derived the integrand with respect to every parameter and then tried to substitute the order of integration but in all cases I get a non-elementary integral. Any ideas how to calculate this? Any help would be appreciated.

• maybe limit test with $1/x^q$? That may take care of some cases, but not all – GPerez Jan 12 '14 at 11:37

Carry out the substitution $y=x^{1-q}$ and then use Newton's generalized binomial theorem.
• It leads to a calculation of the integral. From Wikipedia, binomial theorem, $(a+b)^p=\sum_{k=0}^\infty {p\choose k} a^k b^{p-k}$. In your case $a$ is a power of $x$ and hence so is $a^k$. – JPi Jan 12 '14 at 14:32