Integrals involving roots
Integrals involving roots 2314
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This 3 page Class Notes was uploaded by Alejandra D. Rocha on Friday June 10, 2016. The Class Notes belongs to 2314 at University of Texas at El Paso taught by in Summer 2016. Since its upload, it has received 27 views. For similar materials see Calculus 2 in Math at University of Texas at El Paso.
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Date Created: 06/10/16
Integrals involving roots Let’s look at a couple of examples to see another technique that can be used on occasion to help with these integrals. Example 1 Evaluate the following integral. Solution Sometimes when faced with an integral that contains a root we can use the following substitution to simplify the integral into a form that can be easily worked with. So, instead of letting u be the stuff under the radical as we often did in Calculus I we let u be the whole radical. Now, there will be a little more work here since we will also need to know what x is so we can substitute in for that in the numerator and so we can compute the differential, dx. This is easy enough to get however. Just solve the substitution for x as follows, Using this substitution the integral is now, So, sometimes, when an integral contains the root the substitution, can be used to simplify the integral into a form that we can deal with. Let’s take a look at another example real quick. Example 2 Evaluate the following integral. Solution We’ll do the same thing we did in the previous example. Here’s the substitution and the extra work we’ll need to do to get x in terms of u. With this substitution the integral is, This integral can now be done with partial fractions. Setting numerators equal gives, Picking value of u gives the coefficients. The integral is then, So, we’ve seen a nice method to eliminate roots from the integral and put it into a form that we can deal with. Note however, that this won’t always work and sometimes the new integral will be just as difficult to do.
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