I need to do a direct computation of
$\int_{C}\textbf{B}\cdot d\textbf{r} $
where $\textbf{B} = (\frac{\mu_{0}I}{2\pi}\frac{-y}{x^2+y^2},\frac{\mu_{0}I}{2\pi}\frac{x}{x^2+y^2},0)$ and is a magnetic field created by a electric current $I$ through origin. $C$ is a circle in $xy$-plane with center at $(x_{0},y_{0}) \neq (0,0)$ and radius $a$.
I know the line $\textbf{integral is supposed to be zero}$. But i have a hard time proving it.
Ive tried to change to polar coordinates $x = x_{0} + acos(t)$ and $y = y_{0} + asin(t)$
The expression i get is $\frac{\mu_{0}I}{2\pi} \int^{2pi}_{0} \frac{(y_{0} + asin(t) + x_{0} + acos(t) + a^2)}{(x_{0} + acos(t))^2 + (y_{0} + asin(t))^2} dt$ which is kinda hard to integrate.
I got a tips that i should just use polar coordinates centered at origin and try to argue topologically that the integral is zero which i have no clue how to do.
Any tips will be useful thanks!