  We generalize an integral which arises in several areas in probability and
statistics and which is at the core of the field of distance correlation, a
concept developed by Sz\'ekely, Rizzo and Bakirov (2007) to measure dependence
between random variables. Let $m$ be a positive integer and let ${\cos_m}(u)$,
$u \in \mathbb{R}$, be the truncated Maclaurin expansion of ${\cos}(u)$, where
the expansion is truncated at the $m$th summand. For $t, x \in \mathbb{R}^d$,
let $\langle t,x\rangle$ and $\|x\|$ denote the standard Euclidean inner
product and norm, respectively. We establish the integral formula: For $\alpha
\in \mathbb{C}$ and $x \in \mathbb{R}^d$, $\int_{{\mathbb{R}}^d}
[\cos_m(\langle t,x\rangle) - \cos(\langle t,x\rangle)] \,{\rm
d}t/{\|t\|^{d+\alpha}} = C(d,\alpha) \, \|x\|^{\alpha}$, with absolute
convergence if and only if $2(m-1) < \Re(\alpha) < 2m$. Moreover, the constant
$C(d,\alpha)$ does not depend on $m$.
