Chapter 6.4, Problem 38E

To determine

To evaluate the given integral.

Answer to Problem 38E

  ${\displaystyle \underset{0}{\overset{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{\int }}4\sec x\tan xdx}=4$

Explanation of Solution

Given:

  ${\displaystyle \underset{0}{\overset{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{\int }}4\sec x\tan xdx}$

Calculation:

  $\begin{array}{l}{\displaystyle \underset{0}{\overset{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{\int }}4\sec x\tan xdx}=4{\left[\sec x\right]}_0^{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\\ =4\left[\sec \left(\frac{\pi }{3}\right)-\sec \left(0\right)\right]\\ =4\left[2-1\right]\\ =4\end{array}$