One well-known trick is a way to evaluate the Gaussian integral $G = \int_\mathbb{R} e^{-x^2}dx = \sqrt{\pi}$ by writing
$$G^2 = \left(\int_\mathbb{R} e^{-x^2}dx\right)\left(\int_\mathbb{R} e^{-y^2}dy\right)
= \int_{\mathbb{R}^2} e^{-(x^2+y^2)}dxdy$$
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: Dawson, Robert J. MacG.  On a "singular" integration technique of Poisson.  American Mathematical Monthly **112** (2005), 270-272.

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