Question

What is the lim as x approaches pi [integral 1+tan t from pi to x]/(pi sin x)

Answer 1

Apply l'Hopital's rule:

`\displaystyle\lim_(x\to\pi)(\displaystyle\int_\pi^x (1+\tan(t))\,\mathrm dt)/(\pi\sin(x))=\lim_(x\to\pi)(1+\tan(x))/(\pi\cos(x))=(1+\tan(\pi))/(\pi\cos(\pi))=\boxed{-\frac1\pi}`

where

`\displaystyle(\mathrm d)/(\mathrm dx)\left[\int_\pi^x(1+\tan(t))\,\mathrm dt\right]=1+\tan(x)`

follows from the fundamental theorem of calculus.