Question

The integral $\underset{\frac{\pi }{4}}{\overset{\frac{3\pi }{4}}{\int }}\frac{dx}{1+\cos x}$ is equal to:

• A. $-2$
• B. $2$
• C. $4$
• D. $-1$

Answer 1

The correct option is B $2$

$I=\underset{\frac{\pi }{4}}{\overset{\frac{3\pi }{4}}{\int }}\frac{dx}{1+\cos x}$
$\Rightarrow I=\underset{\frac{\pi }{4}}{\overset{\frac{3\pi }{4}}{\int }}\frac{1-\cos x}{{\sin }^{2}x}dx=\underset{\frac{\pi }{4}}{\overset{\frac{3\pi }{4}}{\int }}({\text{cosec}}^{2}x-\text{cosec}x\cot x)dx$
$\Rightarrow I=[-\cot x+\text{cosec}x{]}_{\frac{\pi }{4}}^{\frac{3\pi }{4}}$
$\Rightarrow I=2$