The value of int_{0}^{frac{pi}{2}} ln left(fr | vedclass

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(B) Let $I = \int_{0}^{\frac{\pi}{2}} \ln \left(\frac{4+3 \sin x}{4+3 \cos x}\right) d x$ ..... $(1)$Using the property $\int_{0}^{a} f(x) d x = \int_

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(B) Let $I = \int_{0}^{\frac{\pi}{2}} \ln \left(\frac{4+3 \sin x}{4+3 \cos x}\right) d x$ ..... $(1)$Using the property $\int_{0}^{a} f(x) d x = \int_{0}^{a} f(a-x) d x$,we get:$I = \int_{0}^{\frac{\pi}{2}} \ln \left(\frac{4+3 \sin(\frac{\pi}{2}-x)}{4+3 \cos(\frac{\pi}{2}-x)}\right) d x$$I = \int_{0}^{\frac{\pi}{2}} \ln \left(\frac{4+3 \cos x}{4+3 \sin x}\right) d x$ ..... $(2)$Adding $(1)$ and $(2)$:$2I = \int_{0}^{\frac{\pi}{2}} \left[ \ln \left(\frac{4+3 \sin x}{4+3 \cos x}\right) + \ln \left(\frac{4+3 \cos x}{4+3 \sin x}\right) \right] d x$$2I = \int_{0}^{\frac{\pi}{2}} \ln \left( \frac{4+3 \sin x}{4+3 \cos x} \cdot \frac{4+3 \cos x}{4+3 \sin x} \right) d x$$2I = \int_{0}^{\frac{\pi}{2}} \ln(1) d x$Since $\ln(1) = 0$,we have $2I = \int_{0}^{\frac{\pi}{2}} 0 d x = 0$.Therefore,$I = 0$.

The value of $\int_{0}^{\frac{\pi}{2}} \ln \left(\frac{4+3 \sin x}{4+3 \cos x}\right) d x$ is

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