I am given the potential:
$$\phi (x, y) = \frac{k}{2}(x^2+y^2) + axy$$
Where a is a constant.
I have to compute:
$$\int_{-\infty}^\infty \int_{-\infty}^\infty \mathrm e^{-\beta\phi}dxdy$$
Changing to polar coordinates:
$$\int_{0}^{2\pi}\int_{0}^{b} e^{-\beta\phi}rdrd\theta$$
What I have done is:
$$\iint e^{-\beta\phi}rdrd\theta = \iint e^{-\frac{\beta Kr^2}{2}} e^{-\beta a r^2 cos\theta sin\theta}rdrd\theta = \iint e^{-\beta f(r, \theta)}rdrd\theta$$
Where
$$f(r, \theta) = \frac{Kr^2}{2} [(1+\frac{cos \theta sin \theta}{K})^2 -(\frac{cos \theta sin \theta}{K})^2]$$
Now I guess I have to do a substitution, defining a new variable. But I do not get the right one, as I do not get a simpler form.
EDIT
Using polar coordinates here is not the best method.
Take squares and try:
$$x = u + v$$
$$y = u - v$$