Trig Substitutions

Trig Substitutions - Trig Substitutions As we have done in...

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Trig Substitutions As we have done in the last couple of sections, let’s start off with a couple of integrals that we should already be able to do with a standard substitution. Both of these used the substitution and at this point should be pretty easy for you to do. However, let’s take a look at the following integral. Example 1 Evaluate the following integral. Solution In this case the substitution will not work and so we’re going to have to do something different for this integral. It would be nice if we could get rid of the square root somehow. The following substitution will do that for us. Do not worry about where this came from at this point. As we work the problem you will see that it works and that if we have a similar type of square root in the problem we can always use a similar substitution. Before we actually do the substitution however let’s verify the claim that this will allow us to get rid of the square root.
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To get rid of the square root all we need to do is recall the relationship, Using this fact the square root becomes, Note the presence of the absolute value bars there. These are important. Recall that There should always be absolute value bars at this stage. If we knew that was always positive or always negative we could eliminate the absolute value bars using, Without limits we won’t be able to determine if is positive or negative, however, we will need to eliminate them in order to do the integral. Therefore, since we are doing an indefinite integral we will assume that will be positive and so we can drop the absolute value bars. This gives, So, we were able to eliminate the square root using this substitution. Let’s now do the substitution and see what we get. In doing the substitution don’t forget that we'll also need to substitute for the dx . This is easy enough to get from the substitution.
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With this substitution we were able to reduce the given integral to an integral involving trig functions and we saw how to do these problems in the previous section . Let’s finish the integral. So, we’ve got an answer for the integral. Unfortunately the answer isn’t given in x ’s as it should be. So, we need to write our answer in terms of x . We can do this with some right triangle trig. From our original substitution we have,
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This note was uploaded on 11/10/2011 for the course MATH 136 taught by Professor Prellis during the Fall '08 term at Rutgers.

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Trig Substitutions - Trig Substitutions As we have done in...

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