What is the integral of tan(x)?

Feb 20, 2016

$\int \tan x \text{d"x = -ln|cosx| + "constant}$

Explanation:

From the chain rule, we know that

$\frac{\text{d"}{"d} x}{\ln \left(f \left(x\right)\right)} = \frac{1}{f \left(x\right)} \cdot f ' \left(x\right)$

Therefore,

$\int \frac{f ' \left(x\right)}{f \left(x\right)} \text{d"x = ln|f(x)| + "Constant}$

We also know that

$\frac{\text{d"}{"d} x}{\cos x} = - \sin x$

And that

$\tan x = \frac{\sin x}{\cos x}$

$= - \frac{- \sin x}{\cos x}$

$= - \frac{\frac{\text{d"}{"d} x}{\cos x}}{\cos x}$

Hence,

$\int \tan x \text{d"x = -ln|cosx| + "constant}$