I have always been curious for how to find this integral, since I can definitely find $\int \frac1{1+x^2}dx$, $\int \frac1{1+x^3}dx$, and so on. But I’m not sure how to approach doing so for any positive real number $n$:
$$\int \frac1{1+x^n} dx$$
I have a couple of potential approaches:
- Use $\lim_\limits{b\to\infty}\int_\limits{0}^b \frac1{1+x^n} dx = \frac{\pi/n}{\sin{(\pi/n)}}$. This can be derived by rewriting the definite integral in terms of the Beta function. What I would do use the implication that $\lim_\limits{b\to\infty}\int_\limits{0}^b \frac1{1+x^n}dx - \frac{\pi/n}{\sin{(\pi/n)}} = 0$ and use some epsilon-delta argument from there.
- Use summations. $$\text{(1)}\,\,\,\, \frac1{1+x^n} = \sum_{k=0}^\infty (-x^n)^k, |x|<1 \\ \text{(2)} \,\,\,\, 1+x^n=(1+x)(1+x+x^2+x^3+…+x^{n-1}), n \in \mathbb{N} $$ But (1) has the issue of its convergence condition, and (2) only holds for natural numbers. Maybe they could be used as the foundation for some argument for a more general case, but as of now I’m not sure how.
