7.2 definite integral — ગણિત ધોરણ 12 સાયન્સ — Question

$\int\limits_{7\pi /4}^{7\pi /3} {\tan xdx}  $ 

$=  - \left. {\log \cos x} \right|_{7\pi /4}^{7\pi /3}$

$=-\left[\log \cos \frac{7 \pi}{3}-\log \cos \frac{7 \pi}{4}\right]$

$=\log \cos \frac{7 \pi}{4}-\log \cos \frac{7 \pi}{3}$

$=\log \left[\frac{\cos \frac{7 \pi}{4}}{\cos \frac{7 \pi}{3}}\right]$ 

$=\log \left[\frac{\cos \left(2 \pi-\frac{\pi}{4}\right)}{\cos \left(2 \pi+\frac{\pi}{3}\right)}\right]$

$=\log \left(\frac{\cos \frac{\pi}{4}}{\cos \frac{\pi}{3}}\right)$ 

$=\log \left(\frac{\frac{1}{\sqrt{2}}}{\frac{1}{2}}\right)$

$=\log \left(\frac{2}{\sqrt{2}}\right)=\log \sqrt{2}$