Is Fubini's theorem the right way to solve this?

Question

I was given this integral: $$\int_{0}^1 \int_{0}^\infty \frac{y\arctan(xy)}{(1+y^2)(1+x^2y^2)} dy dx $$

I tried using Fubini to change the integration order and what I got was:

$$\int_{0}^\infty \frac{(\arctan(y^2) - \arctan(0))}{2(1+y^2)}dy $$ and I cannot integrate further, is this how I should have solved this integral?

Answer 1

The second integral should have the square outside the arctan. And $\arctan(0)$ is $0$, so your answer is $$ \frac{1}{2}\int_0^\infty \frac{\arctan(y)^2}{1+y^2}dy = \frac{1}{6}(\arctan(\infty)^3-\arctan(0)^3) = \frac{1}{6}\left(\frac{\pi}{2}\right)^3 $$