Substitution Rule for Indefinite Integrals

Substitution Rule for Indefinite Integrals - Substitution...

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Substitution Rule for Indefinite Integrals After the last section we now know how to do the following integrals. However, we can’t do the following integrals. All of these look considerably more difficult than the first set. However, they aren’t too bad once you see how to do them. Let’s start with the first one. In this case let’s notice that if we let
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and we compute the differential (you remember how to compute these right?) for this we get, Now, let’s go back to our integral and notice that we can eliminate every x that exists in the integral and write the integral completely in terms of u using both the definition of u and its differential. In the process of doing this we’ve taken an integral that looked very difficult and with a quick substitution we were able to rewrite the integral into a very simple integral that we can do. Evaluating the integral gives,
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As always we can check our answer with a quick derivative if we’d like to and don’t forget to “back substitute” and get the integral back into terms of the original variable. What we’ve done in the work above is called the Substitution Rule . Here is the substitution rule in general. Substitution Rule A natural question at this stage is how to identify the correct substitution. Unfortunately, the answer is it depends on the integral. However, there is a general rule of thumb that will work for many of the integrals that we’re going to be running across. When faced with an integral we’ll ask ourselves what we know how to integrate. With the integral above we can quickly recognize that we know how to integrate However, we didn’t have just the root we also had stuff in front of the root and (more importantly in this case) stuff under the root. Since we can only integrate roots if there is just an x under the root a good first guess for the substitution is then to make u be the stuff under the root. Another way to think of this is to ask yourself what portion of the integrand has an inside function and can you do the integral with that inside function present. If you can’t then there is a pretty good chance that the inside function will be the substitution. We will have to be careful however. There are times when using this general rule can get us in trouble or overly complicate the problem. We’ll eventually see problems where there are more than one “inside function” and/or integrals that will look very similar and yet use completely different substitutions. The reality is that the only way to really learn how to do substitutions is to just work lots of problems and eventually you’ll start to get a feel for how these work and you’ll find it easier and easier to identify the proper substitutions. Now, with that out of the way we should ask the following question. How, do we
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Substitution Rule for Indefinite Integrals - Substitution...

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