$ \lim_{n\to \infty}\int_{0}^{1}\frac{2nx^{n-1}}{1+x}dx=?$

Question

For $n=1,2,...,$ let $f_n(x)=\frac{2nx^{n-1}}{1+x},x\in[0,1].$ Then $$ \lim_{n\to \infty}\int_{0}^{1}f_n(x)dx=?$$

Here $f_n(1)=n$. So the limit function of $f_n(x)$ is not continuous.

Also I was calculating $$\int_{0}^{1}\frac{x^{n-1}}{1+x}dx$$ but I wasn't able to do that.

Any help is appreciated. Thank you.