Question

Evaluate: ${\int }_0^{\frac{\pi }{2}}\log \left(\frac{4+3\sin x}{4+3\cos x}\right)dx$

Answer 1

$I={\int }_0^{\pi /2}\log \left(\frac{4+3\sin x}{4+3\cos x}\right)dx$ ...(1)

Using $a+b-x$ property $\Rightarrow$

$I={\int }_0^{\pi /2}\log \left(\frac{4+3\cos x}{4+3\sin x}\right)dx$

$I=-{\int }_0^{\pi /2}\log \left(\frac{4+3\sin x}{4+3\cos x}\right)dx$ ...(2)

$\left(1\right)+\left(2\right)\Rightarrow$

$2I=0\Rightarrow I=0$