Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2005 / Paper 1 / 8

The positive integrand factors into three one-variable functions on a rectangular domain. Fubini's theorem therefore gives

In first-quadrant polar coordinates, $x=r\cos \theta$, $y=r\sin \theta$, and the Jacobian determinant contributes $r$. The integrand and area element together become $r^2{e}^{-r^2}\sin \theta {\cos }^2\theta drd\theta$. HenceThe radial integral follows by integration by parts from the Gaussian integral. The isolated undefined ratio at the origin does not affect this convergent double integral.

For fixed $x\ne 0$, the chain rule and the derivative of the hyperbolic secant giveThus the stated antiderivative holds, with its integration constant independent of $y$ but allowed to depend on $x$. At $x=0$ the integrand is zero and the antiderivative is simply constant; the displayed formula with $1/x$ is meant for nonzero $x$.

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.