Primitive of Tangent of a x/Cosine Form

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Theorem

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


Proof

\(\displaystyle \int \tan x \rd x\) \(=\) \(\displaystyle -\ln \size {\cos x}\) Primitive of $\tan x$: Cosine Form
\(\displaystyle \leadsto \ \ \) \(\displaystyle \int \tan a x \rd x\) \(=\) \(\displaystyle \frac 1 a \paren {-\ln \size {\cos a x} } + C\) Primitive of Function of Constant Multiple
\(\displaystyle \) \(=\) \(\displaystyle \frac {-\ln \size {\cos a x} } a + C\) simplifying

$\blacksquare$


Also see


Sources