Mathematics - Definite Integration Question with Solution | TestHub

Let $\mathrm{I}={\int }_{7\pi /4}^{7\pi /3}\sqrt{{\tan }^{2}x}dx$ $$={\int }_{7\pi /4}^{7\pi /3}\tan xdx=-{\log \cos x|}_{7\pi /4}^{7\pi /3}$$ $=-\left[\log \cos \frac{7\pi }{3}-\log \cos \frac{7\pi }{4}\right]$ $=\log \cos \frac{7\pi }{4}-\log \cos \frac{7\pi }{3}$ $=\log \left[\frac{\cos \frac{7\pi }{4}}{\cos \frac{7\pi }{3}}\right]=\log \left[\frac{\cos \left(2\pi -\frac{\pi }{4}\right)}{\cos \left(2\pi +\frac{\pi }{3}\right)}\right]$ $=\log \left(\frac{\cos \frac{\pi }{4}}{\cos \frac{\pi }{3}}\right)=\log \left(\frac{\frac{1}{\sqrt{2}}}{\frac{1}{2}}\right)$ $=\log \left(\frac{2}{\sqrt{2}}\right)=\log \sqrt{2}$.