Chapter 6.4, Problem 36E

To determine

To evaluate the given integral.

Answer to Problem 36E

  ${\displaystyle \underset{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$6$}\right.}{\overset{\raisebox{1ex}{$5\pi $}\!\left/ \!\raisebox{-1ex}{$6$}\right.}{\int }}{\csc }^2\theta d\theta }=2\sqrt{3}$

Explanation of Solution

Given:

  ${\displaystyle \underset{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$6$}\right.}{\overset{\raisebox{1ex}{$5\pi $}\!\left/ \!\raisebox{-1ex}{$6$}\right.}{\int }}{\csc }^2\theta d\theta }$

Calculation:

  $\begin{array}{l}{\displaystyle \underset{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$6$}\right.}{\overset{\raisebox{1ex}{$5\pi $}\!\left/ \!\raisebox{-1ex}{$6$}\right.}{\int }}{\csc }^2\theta d\theta }={\left[-\cot \theta \right]}_{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$6$}\right.}^{\raisebox{1ex}{$5\pi $}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\\ =-\left[\cot \left(\frac{5\pi }{6}\right)-\cot \left(\frac{\pi }{6}\right)\right]\\ =-\left[-\sqrt{3}-\sqrt{3}\right]\\ =2\sqrt{3}\end{array}$