## Integration by Parts

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 givewhich is usually written as

It is important to chooseandthe right way round. If there is anterm, thenis usually chosen to be this power, so that the resulting integralterm 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 thatif 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