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Need solution for RD Sharma maths class 12 chapter 19 Definite Integrals exercise Very short answer type question 15

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Answer: 0

Hint: You must know the integration rules of trigonometric function with its limits

Given: \int_{0}^{\frac{\pi}{2}} \log \left(\frac{3+5 \cos x}{3+5 \sin x}\right) d x            .............(i)

Solution:  \int_{0}^{\frac{\pi}{2}} \log \left(\frac{3+5 \cos x}{3+5 \sin x}\right) d x

Property: \int_{a}^{b} f(x) d x=\int_{a}^{b} f(a+b-x) d x

\mathrm{I}=\int_{0}^{\frac{\pi}{2}} \log \frac{3+5 \cos \left(\frac{\pi}{2}-x\right)}{3+5 \sin \left(\frac{\pi}{2}-x\right)} d x

    \begin{aligned} &=\int_{0}^{\frac{\pi}{2}} \log \frac{3+5 \sin x}{3+5 \cos x} d x \\\\ &=-\int_{0}^{\frac{\pi}{2}} \log \frac{3+5 \cos x}{3+5 \sin x} d x \end{aligned}        ................(ii)

Adding (i) and (ii),

\begin{aligned} &21=\int_{0}^{\frac{\pi}{2}} \log \left(\frac{3+5 \cos x}{3+5 \sin x}\right) d x+\left[-\int_{0}^{\frac{\pi}{2}} \log \frac{3+5 \cos x}{3+5 \sin x} d x\right] \\\\ &21=0 \\\\ &I=0 \end{aligned}


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