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Solutions Section Assignment Problems Chapter Notes Chapter Practice Problems Chapter Practice Problems Solutions Chapter Assignment problems E- Book Notes E-Book Practice Problems E-Book Practice Problems Solutions E-Book Assignment Problems Show Page Help Integration by Parts Lets start off with this section with a couple of integrals that we should already be able to do to get us started. First let’s take a look at the following. So, that was simple enough. Now, let’s take a look at, To do this integral we’ll use the following substitution. Again, simple enough to do provided you remember how to do substitutions. By the way make sure that you can do these kinds of substitutions quickly and easily. From this point on we are going to be doing these kinds tot substitutions in our head. If you have to stop and write these out with every problem you will tint that it will take you significantly longer to do these problems.
Now, let’s look at the integral that we really want to do. If we just had an x by itself or by itself we could do the integral easily enough. But, we don’t have them by themselves, they are instead multiplied together. There is no substitution that we can use on this integral that will allow us to do he integral. So, at this point we don’t have the knowledge to do this integral. To do this integral we will need to use integration by parts so let’s derive the integration by parts formula. We’ll start with the product rule. Now, integrate both sides of this. The left side is easy enough to integrate and we’ll split up the right side Of the integral.
Note that technically we should have had a constant of integration show up on the left side after doing the integration. We can drop it at this point since other constants of integration will be showing up down the road and they would just ND up absorbing this one. Finally, rewrite the formula as follows and we arrive at the integration by parts formula. This is not the easiest formula to use however. So, let’s do a couple of substitutions. Both of these are just the standard Call I substitutions that hopefully you are used to by now, Don’t get excited by the fact that we are using two substitutions here. They will work the same way. Using these substitutions gives us the formula that most people think of as the integration by parts formula.
To use this formula we Will need to identify u and DVD, compute du and v and then use the formula. Note as well that computing v is very easy. All we need to do is integrate DVD. So, let’s take a look at the integral above that we mentioned we wanted to do, Example I Evaluate the following integral. Solution So, on some level, the problem here is the x that is in front of the exponential. If that wasn’t there we could do the integral. Notice as well that in doing integration by parts anything that we choose for u will be differentiated. So, it seems that choosing will be a good choice since upon differentiating the x will drop out. Now that wise chosen u we know that DVD will be everything else that remains.
So, here are the choices four and DVD as well as du and v, The integral is then, Once we have done the last integral in the problem we Will add in the constant Of integration to get our final answer. Next, lets take a look at integration by parts for definite integrals. The integration by parts formula for definite integrals is, Integration by Parts, Definite Integrals Note that the in the first term is just the standard integral evaluation notation that you should be familiar with at this point. All we do is evaluate the term, iv in this case, at b then subtract off the evaluation of the term at a. At some level we don’t really need a formula here because we know that when doing definite integrals all we need to do is do the indefinite integral and then do the evaluation.
Lets take a quick look at a definite integral using integration by parts. Example 2 Evaluate the following integral. This is the same integral that we looked at in the first example so we’ll use the same u and DVD to get, Since we need to be able to do the indefinite integral in order to do the definite integral and doing the definite integral amounts to nothing more than evaluating the indefinite integral at a couple of points we will concentrate on doing indefinite integrals in the rest of this section. In fact, throughout most of this chapter this Will be the case. We Will be doing far more indefinite integrals than definite integrals. Lets take a look at some more examples.
Example 3 Evaluate the following integral, There are two ways to proceed With this example. For many, the first thing that they try is multiplying the cosine through the parenthesis, splitting up the integral and then doing integration by parts on the first integral. While that is a perfectly acceptable way of doing the problem it’s more work than e really need to do. Instead of splitting the integral up let’s instead use the following choices for u and DVD. Notice that we pulled any constants out of the integral when we used the integration by parts formula. We will usually do this in order to simplify the integral a little. Example 4 Evaluate the following integral. For this example we’ll use the following choices for u and DVD.
In this example, unlike the previous examples, the new integral will also require integration by parts. For this second integral we will use the following choices. So, the integral becomes, Be careful With the coefficient on the integral for the second application Of integration by parts. Since the integral is multiplied by we need to make sure that the results of actually doing the integral are also multiplied by . Forgetting to do this is one of the more common mistakes with integration by parts problems. As this last example has shown us, we will sometimes need more than one application of integration by parts to completely evaluate an integral. This is something that will happen so don’t get excited about it when does.
In this next example we need to acknowledge an important point about integration techniques. Some integrals can be done in using several different techniques. That is the case vivid the integral in the next example. Example 5 Evaluate the following integral (a) Using Integration by Parts. (b) using a standard Calculus substitution. (a) Evaluate using Integration by Parts, First notice that there are no trig functions or exponentials in this integral. While a good many integration by parts integrals will involve trig functions and/or exponentials not all of them will so don’t get too locked into the idea of expecting them to show up.