define integral and reference $\int_{0}^{1}\frac{1-x^{N-2}}{1-x^N} dx$

Question

I would like to know if there are references on the integral

$$\int_{0}^{1}\frac{1-x^{N-2}}{1-x^N} \ dx$$

given a natural number $N > 2$.

Answer 1

Gradshteyn and Rhyzhik has a number of integrals similar to the one in your post. There is in fact a slight generalization of your integral, that is entry 3.244.2, which is

$$\int_{0}^{1}\frac{x^{p-1}-x^{q-p-1}}{1-x^{q}}=\frac{\pi}{q}\cot\left(\frac{p\pi}{q}\right) \quad (q>p>0)$$

Letting $p=1$ returns the integral in question.

Answer 2

In general, for any real numbers $p\ge 2$, we have

$\displaystyle \boxed{\int_0^{1} \frac{x^{p-2}-1}{x^p-1} d x=\frac{ \pi}{p} \cot \left(\frac{\pi}{p}\right)}\tag*{} $

Using power series, we can tackle the last integral $J$ as

$\displaystyle \begin{aligned}J&= \int_0^1 \frac{1-x^{p-2} }{1-x^{p}}d x \\ & =\sum_{k=0}^{\infty} \int_0^1\left(1-x^{p-2}\right) x^{pk} d x \\& =\sum_{k=0}^{\infty} \int_0^1\left[x^{p k}-x^{p k+p-2}\right] d x \\& =\sum_{k=0}^{\infty}\left[\frac{1}{pk+1}-\frac{1}{pk+p-1}\right] \\& =\frac{1}{p}\left[\sum_{k=0}^{\infty} \frac{1}{k+\frac{1}{p}}+\frac{1}{(-k-1)+\frac{1}{p}}\right] \\& =\frac{1}{p} \lim _{N \rightarrow+\infty} \sum_{k=-N}^N \frac{1}{k+\frac{1}{p}} \\& =\frac{\pi}{p} \cot \left(\frac{\pi}{p}\right)\end{aligned}\tag*{} $ Hence we can conclude that $\displaystyle \boxed{ \int_0^{1} \frac{x^{p-2}-1}{x^{p}-1} d x=\frac{\pi}{p} \cot \left(\frac{\pi}{p}\right)}\tag*{} $