Natural Logarithm - The Natural Logarithm in Integration

The Natural Logarithm in Integration

The natural logarithm allows simple integration of functions of the form g(x) = f '(x)/f(x): an antiderivative of g(x) is given by ln(|f(x)|). This is the case because of the chain rule and the following fact:

In other words,

and

Here is an example in the case of g(x) = tan(x):

Letting f(x) = cos(x) and f'(x)= – sin(x):

where C is an arbitrary constant of integration.

The natural logarithm can be integrated using integration by parts:

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