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:
“Man is, then, only disguise, falsehood, hypocrisy—both in himself and in regard to others. He does not wish any one to tell him the truth; he avoids telling it to others; and all these dispositions, so removed from justice and reason, have a natural root in his heart.”
—Blaise Pascal (1623–1662)
“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)