Chapter 6.2, Problem 34E

To determine

To evaluate

  $3+2{\displaystyle {\int }_0^{\pi /3}\tan x}dx$

Answer to Problem 34E

  $3+2{\displaystyle {\int }_0^{\pi /3}\tan x}dx=3.38629$

Explanation of Solution

Given:Expression is

  $3+2{\displaystyle {\int }_0^{\pi /3}\tan x}dx$

Calculation:

  $3+2{\displaystyle {\int }_0^{\pi /3}\tan x}dx=3+2{\displaystyle {\int }_0^{\pi /3}\tan x}dx$

  $=3+2{\left[-\ln \left(\cos x\right)\right]}_0^{\pi /3}\left(\because {\displaystyle \int \tan x}dx=-\ln \left(\cos x\right)\right)$

  $=3+2\left[-\ln \left(\cos \left(\pi /3\right)\right)+\ln \left(\cos \left(0\right)\right)\right]$

  $=3+2\left[-\ln \left(\frac{1}{2}\right)+\ln \left(1\right)\right]$

  $=3+2\left[-\ln {\left(2\right)}^{-1}+\ln \left(1\right)\right]$

  $=3+2\left(\ln 2\right)$

  $=3+1.38629$

  $=3.38629$