Question

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

• A. $2$
• B. $-2$
• C. $1/2$
• D. $-1/2$

Answer 1

The Correct Option is A

The correct answer is A:2
\(\int^{\frac{3\pi}{4}}_{\frac{\pi}{4}}\frac{dx}{1+cosx}\)
\(=\int^{\frac{3\pi}{4}}_{\frac{\pi}{4}}\frac{1-cosx}{(1-cosx)(1+cosx)}dx\)
\(=\int^{\frac{3\pi}{4}}_{\frac{\pi}{4}}\frac{1-cosx}{1-cos^2x}dx\)
\(=\int^{\frac{3\pi}{4}}_{\frac{\pi}{4}}\frac{1-cosx}{sin^2x}dx=\int^{\frac{3\pi}{4}}_{\frac{\pi}{4}}\frac{1}{sin^2x}-\int^{\frac{3\pi}{4}}_{\frac{\pi}{4}}\frac{cosx}{sin^2x}dx\)
\(=\int^{\frac{3\pi}{4}}_{\frac{\pi}{4}}cosec^2xdx-\int^{\frac{3\pi}{4}}_{\frac{\pi}{4}}cotx.cosecxdx\)
\(=-cot(\frac{3\pi}{4})+cosec(\frac{3\pi}{4})-(cot(\frac{\pi}{4})-cosec(\frac{\pi}{4}))\)
\(=2\)