Question

The integral $\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{1}{3+2\sin x+\cos x}dx$ is equal to:

• A. ${\tan }^{-1}(2)$
• B. ${\tan }^{-1}(2)-\frac{\pi }{4}$
• C. $\frac{1}{2}{\tan }^{-1}(2)-\frac{\pi }{8}$
• D. $\frac{1}{2}$

Answer 1

Answer: (B)

$I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{dx}{3+2\sin x+\cos x}=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{{\sec }^2\frac{x}{2}\cdot dx}{2{\tan }^2\frac{x}{2}+4\tan \frac{x}{2}+4}$
Put $\tan \frac{x}{2}=t$, so
$I=\underset{0}{\overset{1}{\int }}\frac{dt}{(t+1{)}^2+1}={{\tan }^{-1}(x+1)|}_0^1={\tan }^{-1}2-\frac{\pi }{4}$