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More Substitution Rule

# More Substitution Rule - More Substitution Rule In order to...

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More Substitution Rule In order to allow these pages to be displayed on the web we’ve broken the substitution rule examples into two sections. The previous section contains the introduction to the substitution rule and some fairly basic examples. The examples in this section tend towards the slightly more difficult side. Also, we’ll not be putting quite as much explanation into the solutions here as we did in the previous section. In the first couple of sets of problems in this section the difficulty is not with the actual integration itself, but with the set up for the integration. Most of the integrals are fairly simple and most of the substitutions are fairly simple. The problems arise in getting the integral set up properly for the substitution(s) to be done. Once you see how these are done it’s easy to see what you have to do, but the first time through these can cause problems if you aren’t on the lookout for potential problems. Example 1 Evaluate each of the following integrals. (a) [ Solution ] (b) [ Solution ] (c) [ Solution ] Solution (a) This first integral has two terms in it and both will require the same substitution. This means that we won’t have to do anything special to the integral. One of the more common “mistakes” here is to break the integral up and do a separate substitution on each part. This isn’t really mistake but will definitely increase the amount of work we’ll need to do. So, since both terms in the integral use the same substitution we’ll just do everything as a single integral using the following substitution.

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The integral is then, Often a substitution can be used multiple times in an integral so don’t get excited about that if it happens. Also note that since there was a in front of the whole integral there must also be a in front of the answer from the integral. [ Return to Problems ] (b) This integral is similar to the previous one, but it might not look like it at first glance. Here is the substitution for this problem, We’ll plug the substitution into the problem twice (since there are two cosines) and will only work because there is a sine multiplying everything. Without that sine in front we would not be able to use this substitution. The integral in this case is,
Again, be careful with the minus sign in front of the whole integral. [ Return to Problems ] (c) It should be fairly clear that each term in this integral will use the same substitution, but let’s rewrite things a little to make things really clear. Since each term had an x in it and we’ll need that for the differential we factored that out of both terms to get it into the front. This integral is now very similar to the previous one. Here’s the substitution. The integral is,

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[ Return to Problems ] So, as we’ve seen in the previous set of examples sometimes we can use the same substitution more than once in an integral and doing so will simplify the work.
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More Substitution Rule - More Substitution Rule In order to...

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