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.
One partial answer that forms an entire class of such problems is the use complex analysis for evaluating real integrals. When evaluating path integrals in the complex plane, there are many powerful tools such as the Residue Theorem or Cauchy's Integral Formula.
This is a slight cheat (with respect to the answer), as we are technically converting an integral of a single real variable into an integral with respect to a single complex variable. However, through the lens of $\mathbb{C} \simeq \mathbb{R}^2$ with $z=x + iy$, we can view it as converting our integral into a path integral of a vector field in $\mathbb{R}^2$. Given that it is such a useful technique, it bears mentioning.
See for example this math.Se question where I was able to give a slick answer by using the Residue Theorem. For another example, this pdf gives a whole collection of ways to show that $\int_{-\infty}^\infty e^{-\frac{1}{2}x^2} ~d x = \sqrt{2\pi}$. In particular, the ninth proof uses contour integration (i.e., Cauchy's integral formula) and the eleventh proof uses the Fourier transform $(\mathcal{F} f) (y) = \int_{- \infty}^\infty f(x) e^{-i xy} ~dx$.
The Fresnel Integrals can be solved in a similar to the Gaussian Integral through the use of Euler's complex identity. The common element shared by these integrals is that they become much simpler once written in a transformation with polar coordinates. The takeaway here is that thinking about certain integrals in terms of polar coordinates.
