Integration by parts is a common approach to simplify an integral, it is often useful when two fucntions are multiplied together, the rule is like this: is a funtion, its independent variable is , dependent variable is , is another funtion with as independent variable and as its dependent variable, then we have:
For example, if we want to compute:
here we have and , so
Put everything together, we have
This rule is very helpful if is simpler than and is easy to compute. Where does this rule come from? Here is a way to get it through Product Rule for derivatives:
Integrate both side and rearrange, we will have:
if we assume , so we have , replace the terms with in the equation above, we will have:
This is exactly the integration by parts equation we give at the begin, only with replaced by .