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 Székely, 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 $α\in \mathbb{C}$ and $x \in \mathbb{R}^d$, $\int_{\mathbb{R}^d} [\cos_m(\langle t,x\rangle) - \cos(\langle t,x\rangle)] \,{\rm d}t/{\|t\|^{d+α}} = C(d,α) \, \|x\|^α$, with absolute convergence if and only if $2(m-1) < \Re(α) < 2m$. Moreover, the constant $C(d,α)$ does not depend on $m$.