Evaluate the following integrals: ^ 2 _ - 2 ( 2- 2+ )dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_{\frac{-\pi}{2}}\log\Big(\frac{2-\sin\text{x}}{2+\sin\text{x}}\Big)\text{dx}$

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Answer

Let $\text{I}=\int\limits^{\frac{\pi}{2}}_{\frac{-\pi}{2}}\log\Big(\frac{2-\sin\text{x}}{2+\sin\text{x}}\Big)\text{dx}$
Here, $\text{f(x)}=\log\Big(\frac{2-\sin\text{x}}{2+\sin\text{x}}\Big)$
$\text{f}(-\text{x})=\log\Big(\frac{2-\sin\text{x}}{2+\sin\text{x}}\Big)$
$=\log\Big(\frac{2-\sin\text{x}}{2+\sin\text{x}}\Big)=-\log\Big(\frac{2-\sin\text{x}}{2+\sin\text{x}}\Big)=-\text{f(x)}$
Hence f(x) is an odd function
$\therefore\ \text{I}=0$

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