Primitive of Tangent Function/Secant Form

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Theorem

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

where $\sec x$ is defined.


Proof

\(\ds \int \tan x \rd x\) \(=\) \(\ds -\ln \size {\cos x} + C\) Primitive of $\tan x$: Cosine Form
\(\ds \) \(=\) \(\ds \ln \size {\frac 1 {\cos x} } + C\) Logarithm of Reciprocal
\(\ds \) \(=\) \(\ds \ln \size {\sec x} + C\) Secant is Reciprocal of Cosine

$\blacksquare$


Also see


Sources