Evaluate $\int_{0}^1 x^{p}(\log x)^q dx$ Evaluate 
$$\int_{0}^1 x^{p}(\log x)^q dx$$
for $p \in \mathbb{N}$ and $q \in \mathbb{N}$.

Let $$I(p)=\int_0^1x^p\ dx$$ Therefore your integral is $I^{(q)}(p)$ using differentiation under the integral sign. Thus the $q$th derivative of $\frac{1}{p+1}$ is what you are after.
