Integration by parts

This is a powerful technique for integrating an expression which is the product of two other expressions, one of which you know how to integrate.

It is one of the most useful techniques for integration.

The short way to memorise, which is quite symmetrical, is

∫ u dv = uv - ∫ v du

The longer form, which you may be more familiar with, is

∫ u dv/dx dx = uv - ∫ v du/dx dx

It actually follows directly from the formula for differentiating a product of two expressions:

d/dx (uv) = u dv/dx + v du/dx

Just integrate this and rearrange to get the formula above.

Note that it can often be used iteratively, breaking a problem down into a smaller one. See Integrating powers of sin x for an example of this.