Primitive of Tangent Function

From ProofWiki
Jump to navigation Jump to search

Theorem

$\displaystyle \int \tan x \rd x = -\ln \left \vert {\cos x} \right \vert + C$

where $\cos x \ne 0$.


Corollary

$\displaystyle \int \tan x \ \mathrm d x = \ln \left \vert {\sec x} \right \vert + C$

where $\sec x$ is defined.


Proof

\(\displaystyle \int \tan x \rd x\) \(=\) \(\displaystyle \int \frac {\sin x} {\cos x} \rd x\) Definition of Tangent Function
\(\displaystyle \) \(=\) \(\displaystyle -\int \frac {-\sin x} {\cos x} \rd x\) Multiply by $1 = \dfrac {-1}{-1}$
\(\displaystyle \) \(=\) \(\displaystyle -\int \frac {\left({\cos x}\right)'} {\cos x} \rd x\) Derivative of Cosine Function
\(\displaystyle \) \(=\) \(\displaystyle -\ln \left \vert {\cos x} \right \vert + C\) Primitive of Function under its Derivative

$\blacksquare$


Also see


Sources