Question

\[\int\limits_0^{\pi/2} \log \left( \frac{3 + 5 \cos x}{3 + 5 \sin x} \right) dx .\]

 

Answer 1

\[Let\, I = \int_0^\frac{\pi}{2} \log\left( \frac{3 + 5\cos x}{3 + 5\sin x} \right) d x ................(1)\]
\[ = \int_0^\frac{\pi}{2} log\left[ \frac{3 + 5\cos\left( \frac{\pi}{2} - x \right)}{3 + 5\sin\left( \frac{\pi}{2} - x \right)} \right] dx\]
\[ = \int_0^\frac{\pi}{2} log\left( \frac{3 + 5\sin x}{3 + 5\cos x} \right) dx .................(2)\]
\[\text{Adding (1) and (2)}\]
\[2I = \int_0^\frac{\pi}{2} \left[ \log\left( \frac{3 + 5\cos x}{3 + 5\sin x} \right) + log\left( \frac{3 + 5\sin x}{3 + 5\cos x} \right) \right] d x\]
\[ = \int_0^\frac{\pi}{2} \log\left( \frac{3 + 5\cos x}{3 + 5\sin x} \times \frac{3 + 5\sin x}{3 + 5\cos x} \right) dx\]
\[ = \int_0^\frac{\pi}{2} \log1 dx = 0\]
\[Hence\ I = 0\]