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Question
\[\int\limits_{- \pi/4}^{\pi/4} \left| \tan x \right| dx\]
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Solution
\[\int_\frac{- \pi}{4}^\frac{\pi}{4} \left| \tan x \right| d x\]
\[ = \int_\frac{- \pi}{4}^0 - \tan x dx + \int_0^\frac{\pi}{4} \tan x dx\]
\[ = \left[ \log \left( \cos x \right) \right]_\frac{- \pi}{4}^0 + \left[ - \log \left( \cos x \right) \right]_0^\frac{\pi}{4} \]
\[ = - \log\frac{1}{\sqrt{2}} - \log\frac{1}{\sqrt{2}}\]
\[ = 2\log\sqrt{2}\]
\[ = \log2\]
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