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

Answer: (B)

$2$

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

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

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

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

$=\underset{\frac{\pi }{4}}{\overset{\frac{3\pi }{4}}{\int }}{\csc }^{2}xdx-\underset{\frac{\pi }{4}}{\overset{\frac{3\pi }{4}}{\int }}\cot x\csc xdx$

$=-\cot x+\csc x{|}_{\frac{\pi }{4}}^{\frac{3\pi }{4}}$

use limits

$=-\cot \left(\frac{3\pi }{4}\right)+\csc \left(\frac{3\pi }{4}\right)-[-\cot (\frac{\pi }{4})+\csc (\frac{\pi }{4}\left)\right]$

$=2$