Question

${\int }_{\pi /4}^{3\pi /4}\frac{dx}{1+\cos x}$ is equal to:

• A. $2$
• B. $-2$
• C. $\frac{1}{2}$
• D. $-\frac{1}{2}$

Answer 1

The correct option is C $\frac{1}{2}$

Consider the following question.

$I=\underset{\frac{\pi }{4}}{\overset{\frac{3\pi }{4}}{\int }}\frac{1}{(1+\cos x)}$

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

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

$=\frac{1}{2}{{\left(\tan \frac{x}{2}\right)}_{\frac{\pi }{4}}}^{\frac{3\pi }{4}}$

$=\frac{1}{2}\tan \left(\frac{\frac{3\pi }{4}-\frac{\pi }{4}}{2}\right)$

$=\frac{1}{2}\tan \frac{\pi }{4}$

$=\frac{1}{2}$

Hence, this is the required answer.