Integration by Parts is a special method of integration that is often useful when two functions are multiplied together, but is also helpful in other ways.

You will see plenty of examples soon, but first let us see the rule:

∫u v dx = u∫v dx −∫u' (∫v dx) dx

**u**is the function u(x)**v**is the function v(x)

As a diagram:

Let's get straight into an example, and talk about it after:

So we followed these steps:

- Choose u and v
- Differentiate u: u'
- Integrate v: ∫v dx
- Put u, u' and ∫v dx into:
**u∫v dx −∫u' (∫v dx) dx** - Simplify and solve

In English, to help you remember,

**∫u v dx**becomes:
(u integral v) minus integral of (derivative u, integral v)

Let's try some more examples:

Well, that was a spectacular disaster! It just got more complicated.

Maybe we could choose a different u and v?

The moral of the story: Choose

**u**and**v**carefully!
Choose a

**u**that gets simpler when you differentiate it and a**v**that doesn't get any more complicated when you integrate it.
A helpful rule of thumb is I LATE. Choose

**u**based on which of these comes first:**I**: Inverse trigonometric functions such as sin^{-1}(x), cos^{-1}(x), tan^{-1}(x)**L**: Logarithmic functions such as ln(x), log(x)**A**: Algebraic functions such as x^{2}, x^{3}**T**: Trigonometric functions such as sin(x), cos(x), tan (x)**E**: Exponential functions such as e^{x}, 3^{x}

And here is one last (and tricky) example:

## Footnote: Where Did "Integration by Parts" Come From?

It is based on the Product Rule for Derivatives:

(uv)' = uv' + u'v

Integrate both sides and rearrange:

∫(uv)' dx = ∫uv' dx + ∫u'v dx

uv = ∫uv' dx + ∫u'v dx

∫uv' dx = uv − ∫u'v dx

Some people prefer that last form, but I like to integrate v' so the left side is simple:

∫uv dx = u∫v dx − ∫u'(∫v dx) dx