The value of ∫_0^(π)/2 log≤ft((4 + 3sin x)/(4 + 3cos x))dx is

The correct answer is Option (1) → 0

Let

    \( \displaystyle I = \int_{0}^{\frac{\pi}{2}}\log\left(\frac{4 + 3\sin x}{4 + 3\cos x}\right)dx \)

By the property \(\int_0^a f(x)\,dx = \int_0^a f(a - x)\,dx\), replace x by \(\frac{\pi}{2} - x\); then sin and cos swap:

    \( \displaystyle I = \int_{0}^{\frac{\pi}{2}}\log\left(\frac{4 + 3\cos x}{4 + 3\sin x}\right)dx \)

Since \(\log\frac{q}{p} = -\log\frac{p}{q}\), this integral is −I:

    \( I = -I \)
    \( 2I = 0 \)
    \( I = 0 \)