Question

The value of \[\int\limits_0^{\pi/2} \log\left( \frac{4 + 3 \sin x}{4 + 3 \cos x} \right) dx\] is 

 

Options

• 2

• \[\frac{3}{4}\]

• 0

• −2

Answer 1

0

\[Let\ I = \int_0^\frac{\pi}{2} \log\left( \frac{4 + 3\sin x}{4 + 3\cos x} \right) d x ............(1)\]
\[ = \int_0^\frac{\pi}{2} \log\left[ \frac{4 + 3\sin\left( \frac{\pi}{2} - x \right)}{4 + 3\cos\left( \frac{\pi}{2} - x \right)} \right] dx\]
\[ = \int_0^\frac{\pi}{2} \log\left( \frac{4 + 3 \cos x}{4 + 3\sin x} \right) d x ..............(2)\]
\[\text{Adding (1) and (2)}\]
\[2I = \int_0^\frac{\pi}{2} \left[ \log\left( \frac{4 + 3\sin x}{4 + 3\cos x} \right) + log\left( \frac{4 + 3 \cos x}{4 + 3\sin x} \right) \right] d x \]
\[ = \int_0^\frac{\pi}{2} \log\left( \frac{4 + 3\sin x}{4 + 3\cos x} \times \frac{4 + 3 \cos x}{4 + 3\sin x} \right) d x \]
\[ = \int_0^\frac{\pi}{2} \log1 dx = 0\]
\[Hence\ I = 0 \]