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Antiderivatives by simple partial fractions (nonrepeating linear factors only)

Page history last edited by Phoebe 14 years ago

INTEGRATION BY PARTIAL FRACTIONS        

Phoebe 

 

Integration by partial fractions is a useful method to find the antiderivative of a function with a factorable (or factored) denominator. The approach is a simple yet practical one. To make the integration simpler, we will rewrite the integrand as the sum of fractions and integrate them separately. The same rule that requires a common denominator to add fractions will be used to integrate.


 

WHAT TO LOOK OUT FOR

This form of integration is the easiest one to recognize.

The most recognizable sign is that the integrand is a fraction with factorable (or factored) denominator.

Here are some examples of what you might find.

 

Formula

Formula

Formula


 

HOW TO APPROACH IT

(carefully- factoring can be a ten-headed beast for some of us)

 

Let's take the first example. 

Remember it is Formula

 

For now ignore the integral sign and split the denominator into two fractions that equal the numerator with constant numerators A and B. These constants will always exist for the rational function p(x)/q(x) when p(x) and q(x) are both polynomials AND the degree of p(x) is less than the degree of q(x).

http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/partialfracdirectory/PartialFrac.html 

      Formula

Next cross multiply the left side of the equal sign so that each fraction has the same denominator. 

     Formula

     Since all three fractions have the same denominator, we can simplify this to

     Formula

Now substitute different values for x so that each fraction equals zero. 

     let x=-5

     Formula 

     Formula therefore Formula

     

     next, let x=2

     Formula 

     Formula therefore Formula

Now incorporate your integral sign and the values you have just found, integrate, and manipulate

      Formula 

     Formula

     Formula

     Formula

     Formula


STEPS IN ACTION 

If the numerator also has a variable, just follow the same steps as above. 

Let's take the second example. Remember it's  Formula.

The denominator of this integrand can be factored, so the integral we will work with looks like Formula.

Split

     Formula

Cross multiply

     Formula

Substitute

     let x=1

     Formula

     Formula therefore Formula

     next, let x=-3/2

     Formula

     Formula therefore Formula

Incorporate, integrate, and manipulate

     Formula 

     Formula

     Formula


 

WARNING

This can get a bit tricky if you don't use integration by partial fractions when necessary. For example, on one of Alexis's recent tests she tried to use integration by substitution to evaluate Formula. Needless to say it did not work out very well.

 


APPLETS FOR EXTRA EXPLANATIONS

http://archives.math.utk.edu/visual.calculus/4/partial_fractions.2/index.html 

(This link solves for the values of A and B as a system of equations, but the general idea and order of steps are the same as discussed above)


EXAMPLE PROBLEMS

1. Show Alexis the correct way to evaluate Formula.

2. Evaluate Formula.

3. Evaluate Formula.

4. Which of the following is equal to Formula

     A. Formula

     B. Formula

     C. Formula

     D. Formula

5. Which of the following would require the use of integration by partial fractions? Choose more than one if necessary.

     A. Formula

     B. Formula

     C. Formula

     D. Formula

6. Evaluate Formula.

     A. Formula

     B. Formula

     C. Formula

     D. Formula


ANSWERS 

1.  Formula

2. Formula

3. Formula

4. C

5. A, C- This one is a little tricky. D is not correct because the antiderivative is arctan the squared term. B is also not correct because it requires you to complete the square, which will result in arctan of the squared term again.

Here is a full explanation of B. Number 3 on the applet link will show an explanation of C.

Formula

Formula

6. A


HELPFUL EXTERNAL LINKS

http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/partialfracdirectory/PartialFrac.html

http://en.wikipedia.org/wiki/Partial_fractions_in_integration 

http://www.karlscalculus.org/calc11_5.html 

Comments (3)

kforsyth@... said

at 2:53 pm on Feb 1, 2009

I liked it! You had good, concise explanations for how to solve these, and I liked the humor...the 'ten-headed beast' comment particularly. In terms of your questions, I particularly liked #5 because often my biggest problem is recognizing when to use this method. It was really good how you broke up the text with the equations. The last sentence in the beginning description was a little difficult to comprehend (though this could be because I haven't had all my coffee yet...), but otherwise, well done! ~Katherine

Jesse Choukas-Bradley said

at 4:36 pm on Feb 2, 2009

Great job! this is a concept with a difficult, complex explanation and your explanation was clear and concise. Some of the long integral sequences are daunting, but that is difficult to avoid when explaining partial fractions. I liked your example problem and "warning" for tricky pitfalls in particular. Well done

Jesse Choukas-Bradley said

at 7:18 pm on Feb 2, 2009

In addition, I appreciated your breakdown of the steps of partial fraction integration (what to look for, how to approach it, etc.), and I found your examples to be great demonstrations of the concepts

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