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

Option c is correct

$I = \int\limits_0^{\pi /2} {\log \left[ {\cfrac{{4 + 3\sin x}}{{4 + 3\cos x}}} \right]dx}$

Also, $I = \int\limits_0^{\pi /2} {\log \left[ {\cfrac{{4 + 3\sin \left( {\cfrac{\pi }{2} - x} \right)}}{{4 + 3\cos \left( {\cfrac{\pi }{2} - x} \right)}}} \right]} dx$

$\Rightarrow$ $I = \int\limits_0^{\pi /2} {\log \left[ {\cfrac{{4 + 3\cos x}}{{4 + 3\sin x}}} \right]dx}$ $\Rightarrow$ $I = - \int\limits_0^{\pi /2} {\log \left( {\cfrac{{4 + 3\sin x}}{{4 + 3\cos x}}} \right)} dx$

$\Rightarrow$ $I = - I$ $\Rightarrow$ $2I = 0$ $\Rightarrow$ $I = 0$