I recently discovered Differentiation under the integral sign a.k.a Feynman Integration and I read an article which says it can be substituted for contour integration. Therefore, I am assuming this technique is, indeed, very powerful. I was looking for a list of integrals which are, seemingly, hard but are made easy via this technique.
Thanks a lot!
Note that $$ \int_0^1 x^\alpha\ dx=\frac1{\alpha+1},\qquad\text{for }\ \alpha>-1.\tag1 $$ Differentiating $(1)$ $n$ times yields $$ \int_0^1 \frac{\partial^n}{\partial\alpha^n}\left(x^\alpha\right)\ dx=\color{blue}{\int_0^1 x^\alpha \ln^n x\ dx=\frac{(-1)^n n!}{(\alpha+1)^{n+1}}}, \qquad\text{for }\ n=0,1,2,\ldots\tag2 $$