Question

$\underset{0}{\overset{\frac{\pi }{2}}{\int }}\log \left[\frac{4+3\cos x}{4+3\sin x}\right]dx$ is equal to

• A. $0$
• B. $\log 2$
• C. $1$
• D. $\frac{4}{3}$

Answer 1

The correct option is A $0$

Let $I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\log \left[\frac{4+3\cos x}{4+3\sin x}\right]dx....\left(i\right)$
Using property of integration
$I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\log \left[\frac{4+3\sin x}{4+3\cos x}\right]dx....\left(ii\right)$
Adding equation $\left(i\right)$ and $\left(ii\right)$, we get,
$2I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\log \left[\frac{4+3\cos x}{4+3\sin x}\right]dx+\underset{0}{\overset{\frac{\pi }{2}}{\int }}\log \left[\frac{4+3\sin x}{4+3\cos x}\right]dx$
$2I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\log \{\left[\frac{4+3\cos x}{4+3\sin x}\right]\times \left[\frac{4+3\sin x}{4+3\cos x}\right]\}dx$
$2I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\log 1dx=\underset{0}{\overset{\frac{\pi }{2}}{\int }}0dx=0$
$I=0$