How to integrate with a sum in the numerator and denominator.

Question

I need to integrate $\int_{-\infty}^{\infty}\frac{2xy+y^2}{y^2(y^2+4)}\cdot\frac{1}{1+x^2}dx$. I know that since y is just a constant in this problem, that I can just take the denominator of the first fraction out front of the integral, so essentially I just need to integrate $\int_{-\infty}^{\infty}\frac{2xy+y^2}{1+x^2}dx$. To me, it seems like the right move would be to do a u-subsitution with $u=1+x^2$, then $du=2x\cdot dx$ since I have a 2x in the numerator. However, with the other constants, I'm not sure if this is the right direction to go. Any suggestions?

Answer 1

We have that $$\int\frac{2xy+ y^2}{1+x^2}dx = y\int\frac{2x}{1+x^2}dx + y^2\int\frac{1}{1+x^2}dx$$ The first one is a $u$-substitution ($u = 1+x^2$), giving $\ln(1+x^2)$, and the second one is $\arctan(x)$.