# What is the antiderivative of \sec(x)?

What is the antiderivative of $\mathrm{sec}\left(x\right)$?
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einfachmoipf
$\int \mathrm{sec}xdx$
Multiply by $\frac{\mathrm{sec}\left(x\right)+\mathrm{tan}\left(x\right)}{\mathrm{sec}\left(x\right)+\mathrm{tan}\left(x\right)}$.
And we have:
$\int \frac{\mathrm{sec}\left(x\right)\left(\mathrm{sec}\left(x\right)+\mathrm{tan}\left(x\right)\right)}{\mathrm{sec}\left(x\right)+\mathrm{tan}\left(x\right)}dx=$
$=\int \frac{{\mathrm{sec}}^{2}\left(x\right)+\mathrm{sec}\left(x\right)\mathrm{tan}\left(x\right)}{\mathrm{sec}\left(x\right)+\mathrm{tan}\left(x\right)}dx=$
$=\int \frac{1}{u}du=\mathrm{ln}|u|+C=\mathrm{ln}|\mathrm{sec}\left(x\right)+\mathrm{tan}\left(x\right)|+C$
Karen Robbins