How would I integrate the following integral?

Question

$$ \int_{0}^{1} \frac{dx}{\sqrt[5]{1-x^{5}}} $$

Or any integral of this type.

Answer 1

Let $ n \in\mathbb{N}^{*} : $

Using $ \small\left\lbrace\begin{aligned}u&=x^{n}\\ \mathrm{d}x&=\frac{1}{n}u^{\frac{1}{n}-1}\,\mathrm{d}u\end{aligned}\right. $, we get

\begin{aligned}\int_{0}^{1}{\frac{\mathrm{d}x}{\sqrt[n]{1-x^{n}}}}&=\frac{1}{n}\int_{0}^{1}{u^{\frac{1}{n}-1}\left(1-u\right)^{-\frac{1}{n}}\,\mathrm{d}u}=\frac{1}{n}\beta\left(\frac{1}{n},1-\frac{1}{n}\right)=\frac{\Gamma\left(\frac{1}{n}\right)\Gamma\left(1-\frac{1}{n}\right)}{n\Gamma\left(1\right)}=\frac{\pi}{n\sin{\left(\frac{\pi}{n}\right)}}\end{aligned}

Answer 2

It's possible to write down the antiderivative as some special function (in particular 2F1 hypergeometric function), but you don't need to evaluate the integral to prove it doesn't converge

Hint: it acts like $1/x$

Edit: After the change of integration limits, the integral converges.

So the new hint would be to look into the complex plane.