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:
“Children are incurable romantics. Brimful of romance and tragedy, we whirl through childhood hopelessly in love with our parents. In our epic imagination, we love and are loved with a passion so natural and innocent we may never know its like as adults.”
—Roger Gould (20th century)
“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)