Partial Fractions

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Key: Remember the two basic patterns

You should realize very quickly that partial fraction decomposition has absolutely nothing to do with Calculus. Partial fractions allow you to break apart fractions that are nearly impossible to integrate into smaller fractions that are simple to integrate. So I guess you could say Calculus involves partial fractions, but partial fractions don't involve Calculus. This focuses on partial fractions. Just stick an integral symbol in front of each term to make it focus on Calculus... Partial fraction was one of my worst nightmare in calculus class.

Suppose you have a fraction 1 / 15. 1 / 15 is equal to 1 / (3 * 5) which is equal to 1 / (). Partial fractions allow us to say that

1 3 * (3 + 2)

=

A 3

+

B (3 + 2)

Now if you multiply across by 3 * (3 + 2) you'll get 1 = A * (3 + 2) + B * 3. Now you must solve for the constants A and B. And that's it; the partial fraction decomposition is over.

Granted, it does get slightly more complicated when x and y are involved. But there are a few guidelines to follow that make it very easy.

• Two or more unlike terms that are multiplied together create [however many terms you have, call it Z] Z amount of fractions with a unique constant (usually denoted by A, B, C, D,etc.) in the numerator.

  1 x * (x + 3) * (x + 5)

  =

  A x

  +

  B x + 3

  +

  C x + 5

• Like terms being multiplied together [(x + 1) * (x + 1) for example] create however many like terms you have being multiplied together new fractions. Each new fraction is successively raised to one more than the previous fraction's power until the original power is reached.

  1 (x + 2)

  =

  A x + 2

  +

  B (x + 2)

  +

  C (x + 2)

  Since you end up with a cubed term after decomposing, this rule isn't much help by itself. It helps a great deal when it is mixed with the first rule, however.

It is very important to realize how the two rules could be mixed together. Suppose you have two unlike terms being multiplied together. And each unlike term is raised to a power (ie. like terms being multiplied together). For example: 1 / (x * (x + 1)).

Solving for the constants is easy. Note that you always have an equation of the form

a(x) b(x)

=

A c(x)

+

B d(x)

+ ... etc.

To solve for the constants, simply mutiply across by b(x). That will make ALL the fractions disappear. If the fractions don't go away, you've done something incorrectly. Supposing you are OK so far, now set x equal to values that make some (or one) of the terms shrink to zero. Then you should get a simple equation like 1 = A * 0 + B * 5 + C * 0. Then you can assume from there on that B is equal to 1 / 5. Now solve for a different constant.

Now that you've broken down the original fraction into smaller fractions, and solved for all the resulting constants, put the constants back into the original partial fractions. You should now find that integrating will be more normal.

Probable Question: "My a(x) is not equal to 1."
That's fine. Sometimes you will get things like x - 1 = B * (x - 1) + C * x. But look what transpires if you let x = 1. You get 0 = B * 0 + C * 1. Personally, I find it easier when b(x) is not equal to a constant.

Tip: Integration with partial fraction decomposition usually leads to natural logs and arctans (Tan^-1).