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For $x>0$ and $\alpha >1$, the preceding antiderivative givesThe integrand is nonnegative, so Tonelli's theorem permits reversing the two integrals without first assuming absolute integrability. Put $u=xy$ in the inner integral to obtain the hyperbolic-secant Frullani integral:Here ${\int }_0^{\infty }\mathrm{sech}u\tanh udu=[-\mathrm{sech}u{]}_0^{\infty }=1$. Near zero, cancellation makes the original integrand $O(x)$; at infinity it decays exponentially. The finite result therefore also establishes ordinary convergence of the improper integral.