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:
Read more about this topic: Natural Logarithm
Famous quotes containing the words natural and/or integration:
“All the moral laws are readily translated into natural philosophy, for often we have only to restore the primitive meaning of the words by which they are expressed, or to attend to their literal instead of their metaphorical sense. They are already supernatural philosophy.”
—Henry David Thoreau (1817–1862)
“The more specific idea of evolution now reached is—a change from an indefinite, incoherent homogeneity to a definite, coherent heterogeneity, accompanying the dissipation of motion and integration of matter.”
—Herbert Spencer (1820–1903)