Primitive of Secant of a x/Secant plus Tangent Form

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Theorem

$\ds \int \sec a x \rd x = \frac 1 a \ln \size {\sec a x + \tan a x} + C$

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


Proof

\(\ds \int \sec x \rd x\) \(=\) \(\ds \ln \size {\sec x + \tan x}\) Primitive of $\sec x$: Secant plus Tangent Form
\(\ds \leadsto \ \ \) \(\ds \int \sec a x \rd x\) \(=\) \(\ds \frac 1 a \ln \size {\sec a x + \tan a x} + C\) Primitive of Function of Constant Multiple

$\blacksquare$


Also see


Sources