Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2018 / Paper 1 / 17 / c

Differentiate $x/(1+x^2{)}^n$ and integrate between zero and infinity. Its boundary values vanish for every integer $n\ge 1$, givingSince $x^2/(1+x^2{)}^{n+1}=(1+x^2{)}^{-n}-(1+x^2{)}^{-n-1}$, this becomes $0=E_n-2n(E_n-E_{n+1})$. Hence the recurrence relation isThe starting value is $E_1=[\arctan x{]}_0^{\infty }=\pi /2$. Therefore $E_2=\pi /4$, $E_3=3\pi /16$, andThese are also Wallis integrals after the integration by substitution $x=\tan t$.