We can think of integration by parts as a way to undo the product rule. While integration by substitution lets us find antiderivatives of functions that came from the chain rule, integration by parts lets us find antiderivatives of functions that came from the product rule.
Integration by parts requires learning and applying the integration-by-parts formula. Here's the formula, written in both Leibniz and Lagrange notation:
Why the Formula is True
It's time to be skeptical. Why should
be true? Just because we wrote down an equation doesn't mean it's valid.
Sample Problem
- What should be the derivative of
with respect to x?
- What should be the derivative of
with respect to x?
- What's the derivative of uv?
Answer.
is the family of all functions whose derivative is uv'. If we take the derivative of any function in that family, we get uv'. So it makes sense for the derivative of 
to also be uv'. In symbols,
is the collection of all functions whose derivative is vu'. If we take the derivative of any one of those functions, we get vu', so the derivative of 
should also be vu'. In symbols,
- This is a straightforward application of the product rule. The derivative of uv is
uv' + u'v.
Let's return to the question of why the equation
should be valid. Each side of this equation describes a family of functions. If each side of the equation describes the same family of functions, then the equation is valid.
describes the family of functions with derivative uv'.
The expression
describes the family of functions with derivative
Since u'v and vu' are the same, when we simplify we get
u'v + uv' – vu' = u'v.
This means the expression
also describes the family of functions with derivative u'v.
Since the expressions
and
describe the same family of functions, the equation
is valid.
How to Use the Formula
Hopefully, you're now convinced that
To use this formula we have to
- pick u and v',
- figure out what u ' and v are, and
- stick everything in the formula and simplify.
If we're using the notation
then we pick u and dv, figure out du and v, then apply the formula.
.
,
