integration - Evaluate $lim _{nto infty }nint _1^2 frac{dx}{x^2(1+x^n)}$

Evaluate $$\lim _{n\to \infty }n\int _1^2 \frac{dx}{x^2(1+x^n)}$$

without Taylor expansion.

I tried rewriting as

$$\lim_{n\to \infty} n\int _1^2 \frac{1 + x^n - x^n}{x^2(1+x^n)}dx = \lim_{n\to \infty} n\int _1^2 \frac{dx}{x^2} - \lim_{n\to \infty} n\int _1^2 \frac{x^n}{x^2(1+x^n)}dx$$

The first integral is computable, but I don't know how to continue solving the second one.

The answer is $\ln 2$

Answer

By setting $x=z^{1/n}$ we are left with
$$\lim_{n\to +\infty}\int_{1}^{2^n}\frac{dz}{z(1+z)z^{1/n}}=\lim_{n\to +\infty}\left[O\left(\frac{1}{2^n}\right)+\int_{1}^{+\infty}\left(\frac{1}{z}-\frac{1}{z+1}\right)\frac{dz}{z^{1/n}}\right]$$
then by applying the dominated convergence theorem we get that the answer is $\color{red}{\log 2}$.