Continuing on the path of reversing derivative rules in order to make them useful for integration, we reverse the product rule.
Integration by parts
If where and are functions of , then
This is the integration by parts formula. It is very useful in many integrals involving products of functions, as well as others.
For instance, to treat
we choose and . With these choices, we have and , and we have
Note that the choice of and was critical. Had we chosen the reverse, so that and , the result would have been
The resulting integral is no easier to work with than the original; we might say that this application of integration by parts took us in the wrong direction.
So the choice is important. One general guideline to help us make that choice is, if possible, to choose to be the factor of the integrand which becomes simpler when we differentiate it. In the last example, we see that does not become simpler when we differentiate it: is no simpler than .
An important feature of the integration by parts method is that we often need to apply it more than once. For instance, to integrate
we start by choosing and to get
Note that we still have an integral to take care of, and we do this by applying integration by parts again, with and , which gives us
So, two applications of integration by parts were necessary, owing to the power of in the integrand.
Note that any power of x does become simpler when we differentiate it, so when we see an integral of the form
one of our first thoughts ought to be to consider using integration by parts with . Of course, in order for it to work, we need to be able to write down an antiderivative for .
Use integration by parts to evaluate the integral
Solution: If we let and , then we have and . Using our rule for integration by parts gives
We do not seem to have made much progress.
But if we integrate by parts again with and and hence and , we obtain
We may solve this identity to find the anti-derivative of and obtain
With definite integral
For definite integrals the rule is essentially the same, as long as we keep the endpoints.
Integration by parts for definite integrals
Suppose f and g are differentiable and their derivatives are continuous. Then
This can also be expressed in Leibniz notation.
Examples Set 1: Integration by Parts
Evaluate the following using integration by parts.