Consider the following equality:
$\lim_{n\to\infty}\int_{0}^1f_n(x)dx=\int_{0}^1(\lim_{n\to\infty}f_n(x))dx$
where $f_n(x):=\frac{x^n}{1+x^n}\qquad x\in [0,1]$
Since the sequence $(f_n(x))_{n=1}^{\infty}$ is not uniformly convergent, one cannot use the theorem about integration of uniformly convergent function sequences. So here is my question:
How to show that the equality is true?
I think it is equivalent to show that $\lim_{n\to\infty}\int_{0}^1\frac{1}{1+x^n}dx=1$ Then things boil down to calculating $\int_{0}^1\frac{1}{1+x^n}dx$ for every $n$, which is what I have no idea to do.