Primitive of Secant Function

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Theorem

Secant plus Tangent Form

$\ds \int \sec x \rd x = \ln \size {\sec x + \tan x} + C$

where $\sec x + \tan x \ne 0$.


Tangent plus Angle Form

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


Also presented as

Some sources present this result as the primitive of the reciprocal of the cosine function:

\(\ds \int \dfrac {\d x} {\cos x}\) \(=\) \(\ds \ln \size {\sec x + \tan x} + C\)
\(\ds \int \dfrac {\d x} {\cos x}\) \(=\) \(\ds \ln \size {\map \tan {\frac x 2 + \frac \pi 4} } + C\)


Also see