<s>[INST] Evaluate $ \lim_{ n \to \infty }\int_{0}^{ \infty } \frac{1}{(1+x^3)(1+x^n)} dx$ Evaluate $$\displaystyle \lim_{ n \to \infty }\displaystyle \int_{0}^{ \infty } \frac{1}{(1+x^3)(1+x^n)} dx$$
I tried some substitution like $t=x^3+1$ and $t=x^n+1$ but did not work, and also division $$\lim_{ n \to \infty }\displaystyle \int_{0}^{ \infty } \frac{1}{(1+x^3)(1+x^3)(x^{n-3}+1-x^{n-1})} dx$$ but got stuck when I tried partial fraction decomposition.
 [/INST] $$I=\lim_{n \to \infty}\int_{0}^{\infty} \frac{1}{(1+x^3)(1+x^n)}dx=\int_{0}^{1} \frac{1}{1+x^3} dx+\int_1^{\infty} 0~ dx$$
As $\displaystyle \lim_{n \to \infty} x^n =0, 0<x<1$; $\displaystyle\lim_{n \to \infty} x^n= \infty ,x>1.$
$$I=\int_{0}^{1}\left( \frac{1}{3(1+x)}-\frac{2x-1-3}{6(1-x+x^2)}\right) dx$$ $$I=\left[\frac{1}{3} \ln (1+x) -\frac{1}{6} \ln(x^2-x+1)+\frac{1}{\sqrt{3}}\tan^{-1}(2x-1)/\sqrt{3})\right]_{0}^{1}$$
$$I=\frac{1}{3}\ln 2+\frac{\pi}{3\sqrt{3}}$$
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