Statistics How To

How to take the Integral of Natural Log

Calculus >

An integral, also called an antiderivative, is a function F(x) whose derivative is the function f(x). Several rules, derived from the rules of differentiation, make taking derivatives a snap. The general rule for the integral of natural log is ∫ ln(x)dx = x ∙ ln(x) – x + C. So if you are lucky enough to have the simple function ln(x), then the integral of natural log is x ∙ ln(x) – x + C. However, you’ll often be given more complicated functions to deal with. Natural logs are the inverse of ex, but you can also think of natural logs as the time you need to reach a certain level of growth. For example, if your investment is showing 100% growth and you want to know when it will be ten times its size, you’d have to wait ln(10), or 2.302 years — assuming you have continuous compounding.

Step 1: Check the following list for integration rules for more complicated integral of natural log rules. If you find your function there, follow the rule:
natural-log


Step 2: Figure out if you have an equation that is the product of two functions. For example, ln(x)*ex. If that’s the case, you won’t be able to take the integral of the natural log on its own, you’ll need to use integration by parts. (Click the link to read an article on how to integrate using parts).
That’s it!

Tip: Sometimes you’ll have an integral with a natural log that you at first won’t recognize as a product of two functions, like lnx. However, remember that you can rewrite division as multiplication. In this example, lnx can be rewritten as 1x*ln.

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How to take the Integral of Natural Log was last modified: October 15th, 2017 by Stephanie Glen

2 thoughts on “How to take the Integral of Natural Log

  1. Kal

    Please insert comment on how to use U-substitution in the integration of the natural log (Ln).