Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2012 / 6 / a

The tangent half-angle substitution $u=\tan (t/2)$ maps $0\le t<\pi$ to $0\le u<\infty$, withBecause $a>1$, both $a+1$ and $a-1$ are positive. ThusEvaluating the improper integral givesAlthough the change of variable has an infinite endpoint, the original integrand is continuous on the closed interval, and the transformed integrand decays as $u^{-2}$, so the limiting substitution is justified.