how to find this line integral and what is its answer evaluate the line integral $$\int_C (xy^2 dy-x^2y dx), $$ taken in the counter-clockwise sense along the cardioid $$r= a(1+\cos\theta)$$
here putting the parametric form of cardioid $x=a(2\cos t-\cos2t), y= a(2\sin t-\sin2t) $ and taking $\theta$ , $0 $ to $2\pi $ tried  to solve but then it became complicated, so is there any other method and in this method please solve these as well ?

You can use Green's theorem to simplify matters. Indeed,
$$\int_C(xy^2dy-x^2ydx)=\iint(y^2+x^2)dxdy=\int_0^{2\pi}\int_{0}^{a(1+\cos\theta)}r^3drd\theta.$$
It now turns into a fairly straightforward double integral.
