int_{ - pi /2}^{pi /2} {log left( {frac{{2 - | vedclass

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(A) Let $f(	heta) = \log \left( \frac{2 - \sin 	heta}{2 + \sin 	heta} ight)$.We check if the function is odd or even by evaluating $f(-	heta)$:$

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(A) Let $f(	heta) = \log \left( \frac{2 - \sin 	heta}{2 + \sin 	heta} ight)$.We check if the function is odd or even by evaluating $f(-	heta)$:$f(-	heta) = \log \left( \frac{2 - \sin(-	heta)}{2 + \sin(-	heta)} ight) = \log \left( \frac{2 + \sin 	heta}{2 - \sin 	heta} ight)$.Since $\log \left( \frac{2 + \sin 	heta}{2 - \sin 	heta} ight) = \log \left( \left( \frac{2 - \sin 	heta}{2 + \sin 	heta} ight)^{-1} ight) = -\log \left( \frac{2 - \sin 	heta}{2 + \sin 	heta} ight) = -f(	heta)$.Thus,$f(	heta)$ is an odd function.By the property of definite integrals,if $f(x)$ is an odd function,then $\int_{-a}^{a} f(x) dx = 0$.Therefore,$\int_{-\pi/2}^{\pi/2} \log \left( \frac{2 - \sin 	heta}{2 + \sin 	heta} ight) d	heta = 0$.

$\int_{ - \pi /2}^{\pi /2} {\log \left( {\frac{{2 - \sin \theta }}{{2 + \sin \theta }}} \right)\,d\theta = } $

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