  The Cauchy-Schl\"omilch transformation states that for a function $f$ and $a,
\, b > 0$, the integral of $f(x^{2})$ and $af((ax-bx^{-1})^{2}$ over the
interval $[0, \infty)$ are the same. This elementary result is used to evaluate
many non-elementary definite integrals, most of which cannot be obtained by
symbolic packages. Applications to probability distributions is also given.
