Integrals Involving Quadratics

Integrals Involving Quadratics - Integrals Involving...

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Integrals Involving Quadratics To this point we’ve seen quite a few integrals that involve quadratics. A couple of examples are, We also saw that integrals involving , and could be done with a trig substitution. Notice however that all of these integrals were missing an x term. They all consist of a quadratic term and a constant. Some integrals involving general quadratics are easy enough to do. For instance, the following integral can be done with a quick substitution. Some integrals with quadratics can be done with partial fractions. For instance,
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Unfortunately, these methods won’t work on a lot of integrals. A simple substitution will only work if the numerator is a constant multiple of the derivative of the denominator and partial fractions will only work if the denominator can be factored. This section is how to deal with integrals involving quadratics when the techniques that we’ve looked at to this point simply won’t work. Back in the Trig Substitution section we saw how to deal with square roots that had a general quadratic in them. Let’s take a quick look at another one like that since the idea involved in doing that kind of integral is exactly what we are going to need for the other integrals in this section. Example 1 Evaluate the following integral. Solution Recall from the Trig Substitution section that in order to do a trig substitution here we first needed to complete the square on the quadratic. This gives, After completing the square the integral becomes, Upon doing this we can identify the trig substitution that we need. Here it is,
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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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Integrals Involving Quadratics - Integrals Involving...

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