I'm just getting started on double integrals, and I recently saw the super cool way to use double integrals to arrive at
$$\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}$$
So, I am wondering if there are any other integrals involving functions with no elementary antiderivative (for example $\sin(x^2)$ or $\tan(x^3)$) which are readily solved using multiple integrals.