Question:
$\int_{0}^{\frac{\pi}{4}} \tan x d x$
Solution:
Let$I=\int_{0}^{\frac{\pi}{4}} \tan x d x$
$\int \tan x d x=-\log |\cos x|=\mathrm{F}(x)$
By second fundamental theorem of calculus, we obtain
$I=\mathrm{F}\left(\frac{\pi}{4}\right)-\mathrm{F}(0)$
$=-\log \left|\cos \frac{\pi}{4}\right|+\log |\cos 0|$
$=-\log \left|\frac{1}{\sqrt{2}}\right|+\log |1|$
$=-\log (2)^{-\frac{1}{2}}$
$=\frac{1}{2} \log 2$
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