17.7 Rational Functions

In this section we present a few rules for finding antiderivatives of simple rational functions.

To antidifferentiate where is a polynomial, make the substitution .

Let . Then so , and

To find where and is a polynomial of degree less than .

We will find numbers and such that

Suppose (17.57) were valid. If we multiply both sides by we
get

Now take the limit as goes to to get

The reason I took a limit here, instead of saying ``now for we get '' is that is not in the domain of the function we are considering. Similarly

and if we take the limit as goes to , we get

Thus,

I have now shown that if there are numbers and such that (17.57) holds, then (17.58) holds. Since I have not shown that such numbers exist, I will verify directly that (17.58) is valid. Write . Then

Let .

Then

so

and

so

Hence

In this example I did not use formula (17.58), because I find it easier to remember the procedure than the general formula. I do not need to check my answer, because my proof of (17.58) shows that the procedure always works. (In practice, I usually do check the result because I am likely to make an arithmetic error.)

To find where is a polynomial of degree , and does not factor as a product of two first degree polynomials.

Complete the square to write

Then , since if then we have factored , and if we can write , and then

and again we get a factorization of . Since , we can write for some , and

Now

Make the substitution to get an antiderivative of the form

The last antiderivative can be found by a trigonometric substitution.

Let

Let , so and . Then

Now let , so , and . Then

Hence,

To find where is a polynomial of degree .

First use long division to write

where is a polynomial, and is a polynomial of degree . Then use one of the methods already discussed.

Hence

Now let . Then , and

Suppose . Then

and if we take the limit of both sides as we get . Also

and if we take the limit as , we get . Thus

Now

so

and thus

I want to find
. Suppose

Then

If we take the limit of both sides as , we get . Also

and if we take the limit of both sides as , we get . Thus

Hence,

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