The Cauchy-Schlomilch transformation

Abstract

The Cauchy-Schlömilch transformation states that for a function $f$ and $a,b>0$, the integral of $f(x^2)$ and $af((ax-b{x}^{-1}{)}^2$ 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.