Mathematics - Definite Integration Question with Solution | TestHub

Let, $I=\underset{\frac{\pi }{4}}{\overset{\frac{3\pi }{4}}{\int }}\frac{dx}{1+\cos x}$

$\Rightarrow I=\underset{\frac{\pi }{4}}{\overset{\frac{3\pi }{4}}{\int }}\frac{dx}{1+2{\cos }^2{\displaystyle \frac{x}{2}}-1}$

$\Rightarrow I=\frac{1}{2}\underset{\frac{\pi }{4}}{\overset{\frac{3\pi }{4}}{\int }}{\sec }^2\frac{x}{2} dx$

$\Rightarrow I={\left[\tan \frac{x}{2}\right]}_{\frac{\pi }{4}}^{\frac{3\pi }{4}}$

$\Rightarrow I=\tan \frac{3\pi }{8}-\tan \frac{\pi }{8}$

$\Rightarrow I=\left(\sqrt{2}+1\right)-\left(\sqrt{2}-1\right)$

$\Rightarrow I=2$.