Theory

Theory 1

The method of integration by parts (abbreviated IBP) is applicable when the integrand is a product for which one factor is easily integrated while the other becomes simpler when differentiated.

Integration by parts

Suppose the integral has this format, for some functions and :

Then the rule says we may convert the integral like this:

This technique comes from the product rule for derivatives:

Now, if we integrate both sides of this equation, we find:

and the IBP rule follows by algebra.

Extra - Full explanation of integration by parts

(1) Product rule for derivatives:

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(2) Integrate both sides:

Definite IBP

Definite version of FTC:

Choosing factors well

IBP is symmetrical. How do we know which factor to choose for and which for ?

Here is a trick: the acronym “LIATE” spells out the order of choices – to the left for and to the right for :