Question

$\int\limits_{0} ^{\pi/2}\frac{\cos\,2x\,dx}{(\sin\,x+\cos\,x)^2}$ is equal to

• A. 0
• B. $\frac{\pi}{4}$
• C. $\frac{\pi}{2}$
• D. none of these.

Answer 1

Answer: (A)

0

Answer 2

$\int\limits_{0}^{\pi /2} \frac{cos\,2x}{\left(sin\,x+cos\,x\right)^{2}} dx$ $=\int\limits_{0}^{\pi /2} \frac{cos^{2}\,x-sin^{2}\,x}{\left(sin\,x+cos\,x\right)^{2}}dx$ $=\int\limits_{0}^{\pi/ 2}\frac{cos\,x-sin\,x}{cos\,x+sin\,x}dx=\left|log \left(cos\,x+sin\,x\right)\right|_{0}^{\pi/ 2}$ $=log\,1-log\,1=0$