Find $p$ and $q$ so that the integral converges

Question

Find all values of $p$ and $q$ so that the below integral converges: $$ I=\int_{0}^{1} x^p \left(\log\frac{1}{x}\right)^q\;\mathrm{d}x $$

I tried and got the solution as:

$q\geq0$ and $p>q-1$

$-1<q<0$ and $p>-1$

Is it correct?

Solution:

After Substituting $x=e^{-y}$ the integral becomes: $$ I=\int_{0}^{\infty} e^{-(p+1)y} y^q\;\mathrm{d}x $$

Answer 1

Hint:

1) $\ln (\frac{1}{x}) =-\ln(x) $

2) make the change of variables $\ln x = y$.

Answer 2

Under the conditions you specified, the value of the integral is (1+p)^(-1-q) Gamma[1+q]. I hope and wish this could help.