Substitution Rule for Definite Integrals

Substitution Rule for Definite Integrals - Substitution...

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Substitution Rule for Definite Integrals We now need to go back and revisit the substitution rule as it applies to definite integrals. At some level there really isn’t a lot to do in this section. Recall that the first step in doing a definite integral is to compute the indefinite integral and that hasn’t changed. We will still compute the indefinite integral first. This means that we already know how to do these. We use the substitution rule to find the indefinite integral and then do the evaluation. There are however, two ways to deal with the evaluation step. One of the ways of doing the evaluation is the probably the most obvious at this point, but also has a point in the process where we can get in trouble if we aren’t paying attention. Let’s work an example illustrating both ways of doing the evaluation step. Example 1 Evaluate the following definite integral. Solution Let’s start off looking at the first way of dealing with the evaluation step. We’ll need to be careful with this method as there is a point in the process where if we aren’t paying attention we’ll get the wrong answer. Solution 1 : We’ll first need to compute the indefinite integral using the substitution rule. Note however, that we will constantly remind ourselves that this is a definite integral by putting the limits on the integral at each step. Without the limits it’s easy to forget that we had a definite integral when we’ve gotten the indefinite integral computed. In this case the substitution is, Plugging this into the integral gives,
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solution method. The limits given here are from the original integral and hence are values of t . We have u ’s in our solution. We can’t plug values of t in for u . Therefore, we will have to go back to t ’s before we do the substitution. This is the standard step in the substitution process, but it is often forgotten when doing definite integrals. Note as well that in this case, if we don’t go back to t ’s we will have a small problem in that one of the evaluations will end up giving us a complex number. So, finishing this problem gives,
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Substitution Rule for Definite Integrals - Substitution...

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