I was attempting to solve this limit $$\lim_{n \to \infty}\int_{0}^ \infty \frac{nx \arctan(x)}{(1+x)(n^2+x^2)}dx $$
After some time I gave up and saw the solution.
The solution involves the Second mean value theorem for integrals which I never heard of.
The solution used the fact that if $f:[0,1]\to \mathbb{R}$ is a continuous function then $\lim_{n \to \infty}\int_0 ^1 \frac{nf(x)}{x^2n^2+1}dx = \frac{f(0) \pi}{2}$ which is proved using the Second mean value theorem for integrals
I want to ask specifically for books that have this theorem and its proof. Because I want to see what theorem this book have that I don't know.
Note $$ I_n=\int_{0}^ \infty \frac{nx \arctan(x)}{(1+x)(n^2+x^2)}dx\overset{x\to nx}=\int_{0}^\infty \frac{nx \arctan(nx)}{(1+nx)(1+x^2)}dx. $$ Using $$ \frac{nx \arctan(nx)}{(1+nx)(1+x^2)}\le\frac{\pi}{2(1+x^2)}$$ and $$ \int_0^\infty \frac{\pi}{2(1+x^2)}dx<\infty $$ one has, by DCT, $$ \lim_{n\to\infty}I_n=\int_{0}^\infty\lim_{n\to\infty}\frac{nx \arctan(nx)}{(1+nx)(1+x^2)}dx=\int_0^\infty\frac{\pi}{2(1+x^2)}dx=\frac{\pi^2}4. $$