Editing Natural logarithm (section)

==The natural logarithm in integration==
The natural logarithm allows simple [[integral|integration]] of functions of the form <math>g(x) = \frac{f'(x)}{f(x)}</math>: an [[antiderivative]] of {{math|''g''(''x'')}} is given by <math>\ln (|f(x)|)</math>.  This is the case because of the [[chain rule]] and the following fact:
<math display="block">\frac{d}{dx}\ln \left| x \right| = \frac{1}{x}, \ \  x \ne 0</math>

In other words, when integrating over an interval of the real line that does not include <math>x=0</math>, then
<math display="block">\int \frac{1}{x} \,dx = \ln|x| + C</math>
where {{mvar|C}} is an [[arbitrary constant of integration]].<ref>For a detailed proof see for instance: George B. Thomas, Jr and Ross L. Finney, ''Calculus and Analytic Geometry'', 5th edition, Addison-Wesley 1979, Section 6-5 pages 305-306.</ref>

Likewise, when the integral is over an interval where <math>f(x) \ne 0</math>, 
:<math display="block">\int { \frac{f'(x)}{f(x)}\,dx} = \ln|f(x)| + C.</math>
For example, consider the integral of <math>\tan (x)</math> over an interval that does not include points where <math>\tan (x)</math> is infinite:
<math display="block">\int \tan x \,dx = \int \frac{\sin x}{\cos x} \,dx = -\int \frac{\frac{d}{dx} \cos x}{\cos x} \,dx = -\ln \left| \cos x \right| + C = \ln \left| \sec x \right| + C.
</math>

The natural logarithm can be integrated using [[integration by parts]]:
<math display="block">\int \ln x \,dx = x \ln x - x + C.</math>

Let:
<math display="block">u = \ln x \Rightarrow du = \frac{dx}{x}</math>
<math display="block">dv = dx  \Rightarrow v = x</math>
then:
<math display="block">
\begin{align}
\int \ln x \,dx & = x \ln x - \int \frac{x}{x} \,dx \\
& = x \ln x - \int 1 \,dx \\
& = x \ln x - x + C
\end{align}
</math>