In every example I saw of integration in polar coordinates the Jacobian determinant is used, not that I have a problem with the Jacobian, but I wondered if there's a simpler way to show this which will also give me some more intuition about the Jacobian.

If I try to simply write the differentials:

\begin{align} x & = r \cos \theta\\ y & = r \sin \theta\\ dx & = dr \cos \theta - r \sin \theta\ d\theta\\ dy & = dr \sin \theta + r \cos \theta\ d\theta\\ \end{align}

In a double integral you integrate $dxdy$, so if I try to plug in the results I'll get something which is not $r d\theta dr$ \begin{align} dxdy & = \left(dr \cos \theta - r \sin \theta\ d\theta \right) \left( dr \sin \theta + r \cos \theta\ d\theta\right)\\ & = dr^2 \cos \theta \sin \theta - r^2 d\theta^2 \cos \theta\ \sin\ \theta + r\ dr\ d\theta\ (\cos^2 \theta\ - \sin^2\theta ) \end{align}

I don't think I can go anywhere from here, I'm not sure if it's just a calculation mistake or the entire logic is bad.

How do I get this right?

Thanx :)