This is a really short and kind of an interesting thing that popped out in my mind.

**Question**. Consider $$\int^\infty_0 \frac{\sin^2(x)}{x^2(x^2+1)}dx$$ I have tried with integration by parts, substitutions and a lot of ways to try to tackle this, in a way that no Laplaces are used or Residue theorem or Feynman's integration or any advanced deep analysis theory. Because there are already videos about it, but since I'm not yet that far into math, I went and gave it some more last thoughts with everything that I've learned. And I thought about the Series Expansion of $\sin^2(x)$, is there a way to use the series expansion of $\sin^2(x)=x^2+\frac{x^4}{3}-\frac{2x^6}{45}-\frac{x^8}{315}+O(x^9)$ in order to integrate this? This is just an idea and I don't even know if it's possible, it was just a curious thought.