The value of $\int\limits^{\frac{\pi}{2}}_{0}log\left(\frac{5+4sinx}{5+4cosx}\right)dx$ is :

0

The correct answer is Option (1) → 0

$I=\int\limits^{\frac{\pi}{2}}_{0}\log\left(\frac{5+4\sin x}{5+4\cos x}\right)dx$  ...(1)

$I=\int\limits^{\frac{\pi}{2}}_{0}\log\left(\frac{5+4\sin(\frac{\pi}{2}-x)}{5+4\cos(\frac{\pi}{2}-x)}\right)dx=\int\limits^{\frac{\pi}{2}}_{0}\log\left(\frac{5+4\cos x}{5+4\sin x}\right)dx$  ...(2)

Adding (1) and (2)

$2I=\int\limits^{\frac{\pi}{2}}_{0}\log\left(\frac{5+4\sin x}{5+4\cos x}×\frac{5+4\cos x}{5+4\sin x}\right)dx$

$2I=\int\limits^{\frac{\pi}{2}}_{0}\log 1dx=\int\limits^{\frac{\pi}{2}}_{0}0dx=0⇒I=0$