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Question
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Solution
\[Let\, I = \int_{- \frac{\pi}{2}}^\frac{\pi}{2} \log\left( \frac{a - \sin\theta}{a + \sin\theta} \right) d \theta\]
\[Here\, f\left( \theta \right) = \log\left( \frac{a - \sin\theta}{a + \sin\theta} \right)\]
\[Consider\, f\left( - \theta \right) = \log\left[ \frac{a - \sin\left( - \theta \right)}{a + \sin\left( - \theta \right)} \right] = - \log\left( \frac{a - \sin\theta}{a + \sin\theta} \right) = - f\left( \theta \right)\]
\[i . e . , f\left( \theta \right) \text{is odd function} . \]
\[\text{Therefore}, I = 0\]
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