Primitive of Tangent Function

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Theorem

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

where $\cos x \ne 0$.


Corollary

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

where $\sec x$ is defined.


Proof

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

$\blacksquare$


Also see


Sources