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Provide solution for RD Sharma maths class 12 chapter 19 Definite Integrals exercise Very short answer type question 18

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

Given: \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \log \left(\frac{a-\sin \theta}{a+\sin \theta}\right) d \theta


Solution:  I=\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \log \left(\frac{a-\sin \theta}{a+\sin \theta}\right) d \theta            .............(i)

Let  f(x)=\log \left(\frac{a-\sin \theta}{a+\sin \theta}\right)

     f(-x)=\log \left(\frac{a-\sin (-\theta)}{a+\sin (-\theta)}\right)

                   =\log \left(\frac{a+\sin \theta}{a-\sin \theta}\right) \quad[\because \sin (-x)=-\sin x]

                   \begin{aligned} &=-\log \left(\frac{a-\sin \theta}{a+\sin \theta}\right) \\ &=-\mathrm{f}(\mathrm{x}) \end{aligned}

Hence, f(x) is an odd function.


Since, \int_{-a}^{a} f(x) d x=0 if f(x) is an odd.


\therefore \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \log \left(\frac{a-\sin \theta}{a+\sin \theta}\right) d \theta=0



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