I've just get stuck with some task of line integral:
$$\int \frac{xdy - ydx}{x^2+y^2}\quad \text{ over} \ x^2+y^2=R^2$$
I understand that I need to use polar coordinates, and I have such thing:
$$x = r \cos\theta\quad dx = -r \sin\theta d\theta$$
$$y = r \sin\theta\quad dy = r \cos\theta d\theta$$
Then I put it in a task example and get: $$\int (r^2 \cos^2\theta+r^2\sin^2\theta)d\theta/r^2 = \int d\theta$$
But from what to what should I integrate and is the result is right?
You're integrating over the entire circle, so the bounds are $0 \le \theta \le 2\pi$. The integral is
$$ \int_0^{2\pi} d\theta = 2\pi$$