Here it goes
∫ from 0 to 2π/3 tan (x/2) dx
this is my last problem i need to know for my test on friday so please help me out
$\displaystyle \int_0^{\frac{2\pi}{3}} \tan\left(\frac{x}{2}\right) \, dx$
$\displaystyle u = \frac{x}{2}$
$\displaystyle du = \frac{1}{2} dx$
substitute and reset the limits of integration ...
$\displaystyle 2 \int_0^{\frac{2\pi}{3}} \tan\left(\frac{x}{2}\right) \, \frac{1}{2}dx$
$\displaystyle 2 \int_0^{\frac{\pi}{3}} \tan(u) \, du$
integrate ...
$\displaystyle 2\left[-\ln|\cos{u}|\right]_0^{\frac{\pi}{3}}$
use the FTC ...
$\displaystyle 2\left[-\ln\left(\frac{1}{2}\right) + \ln(1)\right] = 2\ln(2)$