For suitable functions f the integral
∫0∞∫0∞…∫0∞(f(x1)+f(x2)+…+f(xn))k+1dx1dx2⋯dxn
is equal to
k!1∫0∞tk(∫0∞e−tf(x)dx)ndt.
To see why this is true, start with the last integral and write the n-th power of the inner integral as a multiple integral over variables x1,…,xn. Then change order of integration and evaluate
∫0∞tke−t(f(x1)+…+f(xn))dt=(f(x1)+…+f(xn))k+1Γ(k+1)
which follows from the integral representation
Γ(z)=∫0∞tz−1e−tdt
of the gamma function. Note that we don't need to assume that k is an integer.
I came across this trick while browsing the Journal of Experimental Mathematics and reading the article A Proof of a Recurrence for Bessel Moments. In this article Jonathan M. Borwein and Bruno Salvy are interested in the case where f is the hyperbolic cosine and the integral
K0(t)=∫0∞e−tcosh(x)dx
is a modified Bessel function.