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.

The integrand is dominated by $\frac 1 {1+x^{3}}$ which is integrable. Hence you can take the limit inside and you get $\int_0^{1}\frac  1 {1+x^{3}}dx$.
