The Cauchy–Schlömilch transformation

ABSTRACTThe Cauchy–Schlomilch transformation states that for a function f and a, b>0, the integral of f(x2) and af((ax−bx−1)2) over the interval [0,∞) 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 are also given.