What is the mistake in evaluating $\lim\limits_{n \to \infty} \int_0^ \infty \frac{nx\arctan{x}}{(1+x)(x^2+n^2)}dx$

Question

The answer to this limit: $\lim\limits_{n \to \infty} \int_0^ \infty \frac{nx\arctan{x}}{(1+x)(x^2+n^2)}dx$ is $\frac{\pi^2}{4}$ but one can make this argument: for $x>0$ $$\frac{x}{1+x}<1$$ $$\frac{1}{x^2+n^2} < \frac{1}{n^2}$$ so $\lim\limits_{n \to \infty} \int_0^ \infty \frac{nx\arctan{x}}{(1+x)(x^2+n^2)}dx< \lim\limits_{n \to \infty}\int_0^ \infty \frac{\arctan(x)}{n} =\lim\limits_{n \to \infty}\frac{\pi}{2n} =0$

there should be a mistake somewhere but I couldn't find it.

Answer 1

$\int_0^{\infty} \arctan xdx=\infty$. In fact, there exists $T$ such that $\arctan x >1$ for all $x>T$ so $\int_T^{\infty}\arctan xdx =\infty$.