7.1 Integration by Parts

1. 7.1 Integration by Parts If you remember, we have the product rule in differentiation. However, we haven’t seen an analogy in integration yet. And in fact, MA180 did not teach you any technique to integrate a non-trivial product such as The reason is that these integrals are not easy to evaluate, and some of them are not even possible to integrate. Fortunately, the majority can be evaluated by a special technique called Integration by Parts.

2. Review To understand why the technique is called integration by parts, we should look at the product rule again. We can see that this is actually “differentiation by parts” because we first differentiate part of the product, leaving the other part unchanged, and then come back to differentiate the other part. And if you agree that (symbolic) integration is the reverse of differentiation, then we should use a similar technique to integrate products of functions. Hence we should consider “integration by parts”.

3. Example: Evaluate Clearly, the function xcosx consists of two parts namely, x and cosx. Therefore we can try integrating one of them first. We use subtraction because this should be the reverse of the product rule. We differentiate the answer. How do we check our answer? This is pretty close to what we want, and if we can choose the “something” in such a way that

4. Evaluate Clearly, the function xcosx consists of two parts namely, x and cosx. Therefore we can try integrating one of them first. We use subtraction because this should be the reverse of the product rule. We differentiate the answer. How do we check our answer? This is pretty close to what we want, and if we can choose the “something” in such a way that then we are done.

5. Can you think of “something” that Of course, that “something” should be –cosx ! We know this because and we can choose C = 0 at the moment. Summarizing, we have

6. So far so good, right? Not really, because how do we know in general what that “something” should be? If you look back, you would see that and the “sinx” comes from Do we have a quicker method in general?

7. General case “something” is chosen such that but Hence or equivalently,

8. Summarizing This is of course hard to remember, and hence we have to use a better notation. then g(x) = v’(x) and or if we replace f(x) by u(x), then we have an even easier formula and this can be abbreviated as

9. Now that we know but if we are given How do we know which one should be u(x) or which one should be v’(x)? • Here are the guidelines. • v’(x) should be chosen such that it can be integrated easily, • u(x) should be chosen such that its derivative is simpler or lower in positive degree.