The value of the integral int_{-frac{pi}{4}}^ | vedclass

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Question

(A) Let $I = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \log (\sec 	heta - 	an 	heta) \, d	heta$. Consider the function $f(	heta) = \log (\sec 	heta

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$0$

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(A) Let $I = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \log (\sec 	heta - 	an 	heta) \, d	heta$. Consider the function $f(	heta) = \log (\sec 	heta - 	an 	heta)$. We check if $f(	heta)$ is an odd function by evaluating $f(-	heta)$: $f(-	heta) = \log (\sec(-	heta) - 	an(-	heta)) = \log (\sec 	heta + 	an 	heta)$. Since $\sec 	heta + 	an 	heta = \frac{1}{\sec 	heta - 	an 	heta}$,we have: $f(-	heta) = \log \left( \frac{1}{\sec 	heta - 	an 	heta} ight) = -\log (\sec 	heta - 	an 	heta) = -f(	heta)$. Since $f(	heta)$ is an odd function and the interval $[-\frac{\pi}{4}, \frac{\pi}{4}]$ is symmetric about the origin,the integral of an odd function over this interval is $0$. Therefore,$I = 0$.

The value of the integral $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \log (\sec \theta - \tan \theta) \, d\theta$ is

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