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Definite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integrals2 Marks
Question
Evaluate the following integrals:$\int\limits^\frac{\pi}{2}_{-\frac{\pi}{2}}\log\Big(\frac{\text{a}-\sin\theta}{\text{a}+\sin\theta}\Big)\text{d}\theta$
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Answer

Let $\text{I}=\int\limits^\frac{\pi}{2}_{-\frac{\pi}{2}}\log\Big(\frac{\text{a}-\sin\theta}{\text{a}+\sin\theta}\Big)\text{d}\theta$ Here, $\text{f}(\theta)=\log\Big(\frac{\text{a}-\sin\theta}{\text{a}+\sin\theta}\Big)$ Consider, $\text{f}(-\theta)=\log\bigg[\frac{\text{a}-\sin(-\theta)}{\text{a}+\sin(-\theta)}\bigg]$$=-\log\Big(\frac{\text{a}-\sin\theta}{\text{a}+\sin\theta}\Big)=-\text{f}(\theta)$
i.e., $\text{f}(\theta)$ is odd function. Therefore, $\text{I}=0$

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