One well-known trick is a way to evaluate the Gaussian integral $G = \int_\mathbb{R} e^{-x^2}dx = \sqrt{\pi}$ by writing
$$G^2 = \left(\int_\mathbb{R} e^{-x^2} \, dx\right)\left(\int_\mathbb{R} e^{-y^2} \, dy\right)
= \int_{\mathbb{R}^2} e^{-(x^2+y^2)} \, dx \, dy$$
which when transformed to polar coordinates becomes
$$G^2 = 2\pi \int_0^\infty e^{-r^2} r \, dr = \pi \int_0^\infty e^{-u} \, du = \pi$$
via the substitution $u=r^2$.  It appears this idea is due to Poisson.

In a 2005 note in the American Mathematical MONTHLY, R. Dawson has observed that this is a trick that only works once; there are no other integrals that can be evaluated by this method.  Specifically:

**Theorem.** Any Riemann-integrable function $f$ on $\mathbb{R}$, such that $f(x)f(y) = g(\sqrt{x^2+y^2})$ for some $g$, is of the form $f(x)=ke^{ax^2}$. 

See: <cite authors="Dawson, Robert J. Mac G.">_Dawson, Robert J. Mac G._, [**On a “singular” integration technique of Poisson**](https://dx.doi.org/10.2307/30037446), Am. Math. Mon. 112, No. 3, 270-272 (2005). [ZBL1088.26500](https://zbmath.org/?q=an:1088.26500).</cite>

So if a technique is a trick that works twice, this one is definitely still a trick.