The value of $\int_{–\pi/2}^{\pi/2}{\log{\left( \frac{2 - \sin heta}{2 + \sin heta} ight)d heta}}is$

Question

The value of $\int_{–\pi/2}^{\pi/2}{\log{\left( \frac{2 - \sin	heta}{2 + \sin	heta} ight)d	heta}}is$

Dr. Abubacker · Accepted Answer

(a)
Let $f(	heta) = \log\left( \frac{2 - \sin	heta}{2 + \sin	heta} ight)$
Now, $f( - 	heta = \log\left( \frac{2 + \sin	heta}{2 - \sin	heta} ight)$
$= - \log\left( \frac{2 - \sin	heta}{2 + \sin	heta} ight) = - f(	heta)$
$	herefore f(	heta)$is an odd function.
So, $\int_{- \pi/2}^{\pi/2}{\log{\left( \frac{2 - \sin	heta}{2 + \sin	heta} ight)d	heta = 0}}$

Q. The value of $\int_{–\pi/2}^{\pi/2}{\log{\left( \frac{2 - \sin\theta}{2 + \sin\theta} \right)d\theta}}is$

Explanation

Reviewed by Dr. Abubacker

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