Question

The integral ${\int }_{\frac{\pi }{4}}^{\frac{3\pi }{4}}\frac{dx}{1+\cos x}$

• A. $2$
• B. $4$
• C. $-1$
• D. $-2$

Answer 1

The correct option is A $2$

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

${\int }_{\pi /4}^{3\pi /4}\frac{dx}{2{\cos }^{2}x/2}=\frac{1}{2}{\int }_{\pi /4}^{3\pi /4}{\sec }^{2}x/2dx\frac{x}{2}=t$

$=\frac{1}{2}{\int }_{\pi /4}^{3\pi /4}2{\sec }^{2}tdt$

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

$=\tan \frac{3\pi }{4}+\tan \frac{\pi }{4}$

$=\tan \frac{\pi }{4}+\tan \frac{\pi }{4}$

$=2$