Integration by parts is used to integrate a product. It is derived from the product rule for differentiating a product:
We subtract a term from the right hand side to give
and then integrate to give
which is usually written as
It is important to choose
and
the right way round. If there is an
term, thenis usually chosen to be this power, so that the resulting integral
term is easier to integrate than the original integral. It is useful to write down all the terms of the equation first then it becomes very easy to substitute them all into the formula.
Example: Find
Example:
It is usually the case that
if this is possible but there are exceptions. The best example is when we have to integrate a
Example: Find
Example: Find
To integrate this we have to write
