Primitive of Secant Function/Tangent plus Angle Form

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Theorem

$\ds \int \sec x \rd x = \ln \size {\map \tan {\frac x 2 + \frac \pi 4} } + C$


Proof

\(\ds \int \sec x \rd x\) \(=\) \(\ds \ln \size {\tan x + \sec x} + C\) Primitive of $\sec x$: Secant plus Tangent Form
\(\ds \) \(=\) \(\ds \ln \size {\map \tan {\frac x 2 + \frac \pi 4} } + C\) Tangent of Half Angle plus $\dfrac \pi 4$

$\blacksquare$


Sources