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Please solve RD Sharma Class 12 Chapter 19 Definite Integrals Exercise 19.4 (b) Question 30 maths textbook solution.

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Hints:-  You must know the integral rules of trignometric functions and odd even functions.

Given:-  \int_{-a}^{a} \log \left(\frac{a-\sin \theta}{a+\sin \theta}\right) d \theta, a>0

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

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

f(x) is an odd function

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

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