Partial Fractions

This technique is very appropriate for integrals when you have some mess of a fraction with no hopes of being able to integrate it nicely.

A partial fraction is just taking any fraction and breaking it up into smaller fractions that can later be added together to make the same original fraction.

The first thing you must do to determine if you can do te partial fraction technique is see if your equation is a candidate for it. To do this you check if the degree of the numerator is less than the denominator. If it is not you have yourself an improper fraction and will need to do long polynomial division. Else you can begin doing the partial fraction technique.

Example of an improper fraction

2x^3 + x^2 -10
-----------------  <----degrees are the same
 x^3-3x-2

Now to create your partial fraction:

***Important********

These are the forms your broken up fraction will either be. These are in a sense your formula.


         A                                                   Bx + C
   --------------                                ----------------------
     ( x-a)^k                                    (x^2 + bx + y)^k


Where a , b, y are some values. Where k is some exponent.

1. If the denominator contains linear fractions. Factor them out.

    5                        5
-----------  =   -----------
x^2+x-6        (x-2)(x+3)

2. Use the formulas and break up your equation appropriately. 

  5                        5                             A                                     B
-----------  =   -----------     =      ------------------   +      ---------------------
x^2+x-6        (x-2)(x+3)                   (x-2)                               (x+3)
       
                      

3. Multiple both sides by the left
 
5 = A(x+3) + B(x-2)

4. Takes the roots(the what the x's are) and plug them in to solve for A and B.

x = -3, 2

If x = -3 :    5 = A(0) + B(-5) =  -5B;   B = -1

If x = 2:      5 = A(5) + B(0) = 5A ;   A = 1

5. we have found our A and B. So time to fill in our partial fractions for A and B.

         1                                    -1
------------------   +      ---------------------
     (x-2)                               (x+3)

There you this looks like something you could integrate much easier. Enjoy.