Solving a 2 variable integral with a delta function How can this integrals that include the dirac delta function be solved?
$$\int_{-\infty}^{+\infty}dq\int_{-\infty}^{+\infty}dp\cdot p^n\cdot \delta(p^2+q^2-E)$$

In polar coordinates,
$$\int_{\theta=0}^{2\pi}\int_{r=0}^\infty r^n\cos^n\theta\,\delta(r^2-E)\,r\,dr\,d\theta
=\int_{\theta=0}^{2\pi}\cos^n\theta\,d\theta\int_{r=0}^\infty r^{n+1}\,\frac{\delta(r-\sqrt E)}{2\sqrt E}\,dr\\
=\frac{ E^{n/2}}2\int_{\theta=0}^{2\pi}\cos^n\theta\,d\theta
.$$
We used the fact that $2\sqrt E\,\delta(r^2-E)=\delta(r-\sqrt E)$.
