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) Setup: functions
and are established. Recognize functions
and in the integrand:
(2) Product rule for derivatives.
Using primes notation:
(3)
Integrate both sides of product rule.
Integrate with respect to an input variable labeled ‘
’: Rearrange with algebra:
(4) This is “integration by parts” in final form.
Addendum: definite integration by parts
Definite version of FTC.
Apply FTC to
:
(5) Integrate the derivative product rule using specified bounds.
Perform definite integral on both sides, plug in definite FTC, then rearrange:
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 :