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:
“Your child should feel entitled to cry when you leave; crying is a natural thing for a child to do when she feels bad. The fact that your child cries when you go doesn’t mean she will never like day care. It just means she wants you to stay.”
—Amy Laura Dombro (20th century)
“Look back, to slavery, to suffrage, to integration and one thing is clear. Fashions in bigotry come and go. The right thing lasts.”
—Anna Quindlen (b. 1952)