# Primitive of Tangent Function/Cosine Form

## Theorem

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

where $\cos x \ne 0$.

## Proof

 $\ds \int \tan x \rd x$ $=$ $\ds \int \frac {\sin x} {\cos x} \rd x$ Definition of Real Tangent Function $\ds$ $=$ $\ds -\int \frac {-\sin x} {\cos x} \rd x$ Multiply by $1 = \dfrac {-1} {-1}$ $\ds$ $=$ $\ds -\int \frac {\paren {\cos x}'} {\cos x} \rd x$ Derivative of Cosine Function $\ds$ $=$ $\ds -\ln \size {\cos x} + C$ Primitive of Function under its Derivative

$\blacksquare$