Answer
$-6\ln(\dfrac{\sqrt 3}{3}) \approx 3.2958$
Work Step by Step
Our aim is to solve the integral $ \int_{\pi}^{2\pi} \tan (x/6) \ dx$
Let us consider that $a =\dfrac{x}{6}$ and $\ dx=6 \ da$
Now, $\int_{\pi}^{2\pi} \tan (x/6) \ dx =6 \int \tan a \ da$
or, $=-6 \ln |\cos a| +C$
or, $=-6 \ln |\cos (x/6)| +C$
Now, $\int_{\pi}^{2\pi} \tan (x/6) \ dx=[-6 \ln |\cos (x/6)|]_{\pi}^{2\pi}\\=-6[\ln(\dfrac{1}{2})-\ln(\dfrac{\sqrt 3}{2})]\\=-6\ln(\dfrac{\sqrt 3}{3}) \approx 3.2958$
