Question

\[\int\limits_{- \pi/4}^{\pi/4} \left| \tan x \right| dx\]

Answer 1

\[\int_\frac{- \pi}{4}^\frac{\pi}{4} \left| \tan x \right| d x\]

\[ = \int_\frac{- \pi}{4}^0 - \tan x dx + \int_0^\frac{\pi}{4} \tan x dx\]

\[ = \left[ \log \left( \cos x \right) \right]_\frac{- \pi}{4}^0 + \left[ - \log \left( \cos x \right) \right]_0^\frac{\pi}{4} \]

\[ = - \log\frac{1}{\sqrt{2}} - \log\frac{1}{\sqrt{2}}\]

\[ = 2\log\sqrt{2}\]

\[ = \log2\]