Evaluate the definite integral int_{0}^{frac{ | vedclass

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(A) Let $I = \int_{0}^{\frac{\pi}{4}} 	an x \,dx$. We know that $\int 	an x \,dx = \log |\sec x| = -\log |\cos x|$. Let $F(x) = -\log |\cos x|$. By

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$\frac{1}{2} \log 2$

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(A) Let $I = \int_{0}^{\frac{\pi}{4}} 	an x \,dx$. We know that $\int 	an x \,dx = \log |\sec x| = -\log |\cos x|$. Let $F(x) = -\log |\cos x|$. By the second fundamental theorem of calculus,$I = F\left(\frac{\pi}{4}ight) - F(0)$. $I = [-\log |\cos x|]_{0}^{\frac{\pi}{4}}$. $I = -\log |\cos \frac{\pi}{4}| - (-\log |\cos 0|)$. $I = -\log |\frac{1}{\sqrt{2}}| + \log |1|$. Since $\log 1 = 0$,we have $I = -\log (2^{-\frac{1}{2}}) = -(-\frac{1}{2}) \log 2 = \frac{1}{2} \log 2$.

Evaluate the definite integral $\int_{0}^{\frac{\pi}{4}} \tan x \,dx$.

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