# Evaluate the following as true or false. int tan(2x)dx=-1/2 ln|cos (2x)|+C

Evaluate the following as true or false.
$\int \mathrm{tan}\left(2x\right)dx=-\frac{1}{2}\mathrm{ln}|\mathrm{cos}\left(2x\right)|+C$
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Solution: True
$\int \mathrm{tan}\left(2x\right)dx=-\frac{1}{2}\mathrm{ln}|\mathrm{cos}2x|+C$
$I=\int \mathrm{tan}\left(2x\right)dx$, substitute 2x=t, 2dx=dt
$I=\frac{1}{2}\int \mathrm{tan}\left(t\right)dt$
$I=\frac{1}{2}\mathrm{ln}|\mathrm{sec}x|+c,\mathrm{sec}x=\frac{1}{\mathrm{cos}x}=\left(\mathrm{cos}x{\right)}^{-1},\mathrm{ln}{a}^{m}=m\mathrm{ln}a$
$\therefore I=-\frac{1}{2}\mathrm{ln}|\mathrm{cos}x|+C$