Evaluating the line integral $\int \frac{xy^2}{x^2+y^2}\mathrm dy$ around a circle

Question

Evaluate the line integral

$$\int \frac{xy^2}{x^2+y^2}\mathrm dy$$

Around the circle $x^2+y^2=a^2$ on the positive direction

How to solve this? I don't have an idea. Should I parameterize $x=t$ and then get $y=\sqrt{a^2-t^2}$, find the $\mathrm dt$ and then substituting in the integral?

But what is the bounds? From $0$ to $a$?

Please verify and help me. Thanks

Answer 1

Put $x=a\cos \theta, y=a\sin \theta$. the integral becomes $a^{2}\int_0^{2\pi} \cos^{2} \theta \sin^{2} \theta d\theta$. Can you evaluate this?