Aus Online Mathematik Brückenkurs 2

If we use the definition of \displaystyle \tan x and write the integral as

\displaystyle \int\tan x\,dx = \int\frac{\sin x}{\cos x}\,dx

we see that the numerator \displaystyle \sin x is the derivative of the denominator (apart from the minus sign). Hence, the substitution \displaystyle u=\cos x will work,

\displaystyle \begin{align} \int\frac{\sin x}{\cos x}\,dx &= \left\{\begin{align} u &= \cos x\\[5pt] du &= (\cos x)'\,dx = -\sin x\,dx \end{align}\right\}\\[5pt] &= -\int\frac{du}{u}\\[5pt] &= -\ln |u| + C\\[5pt] &= -\ln |\cos x| + C\,\textrm{.} \end{align}

Note: \displaystyle -\ln \left| \cos x \right|+C is only a primitive function in intervals in which \displaystyle \cos x\ne 0.