Integration by parts is a method for find antiderivatives of a product of functions. It can loosely be thought of as using the product rule "backwards". This rule arises as the product rule creates two terms when used, one of which might be easier to integrate than the other.

The rule can be written either using Lagrange's notation as follows:

Or using Leibniz's notation:

Where , and are all some "group" of algebra, and .

When choosing to use integration by parts the function should have a convenient derivative like , , or and should have a convenient antiderivative like , , or even just a constant .

Definite Integral

To evaluate a definite integral using integration by parts we need to remember to still apply the the fundamental theorem of calculus after finding the antiderivative this can be done as follows:

where

LIATE Rule

The LIATE rule is a general rule of thumb for picking which function will be the and involves choosing the function highest up on the following list.

1. (L)ogarithmic functions - , or .
2. (I)nverse trigonometric functions - ie. , , etc.
3. (A)lgebraic functions - Usually polynomials like , or .
4. (T)rigonometric functions - ie. , , etc.
5. (E)xponential functions - or .

Example

Given the integral , we can see that the first function to appear on the list is the logarithmic function meaning we should set and . We can then use integration by parts to transform this integral.

Examples

Example 1

Evaluate

Example 2 - Repeated Application

Evaluate

Example 3 -

Evaluate

Example 4 - Definite Integral

Evaluate