Educator

Partial Fractions

Main formula: Something like

can be separated into

Hints and tips:

• You can always factor the denominator down into linear (degree one) and quadratic (degree two) factors. If you have factors that are cubic or higher, you should be able to factor more.

• To factor a cubic, test factors of the constant term divided by factors of the leading coefficient as roots. A quick way to check if they work is synthetic substitution (also known as synthetic division).

• Sometimes you can figure out the A and B by plugging in a particular value of x to both sides of the equation. (Choose a value that makes one of the expressions equal to 0.)

• If the degree of the numerator is the same or greater than the degree of the denominator, use long division first. Based on the results of your long division, split your integral into a polynomial part (easy to integrate) and a rational function that you can apply partial fractions to.

• If you have a quadratic in the denominator that doesn’t factor, then follow the following steps in order:

1. Use u = denominator first.

2. Then separate into .

3. On the first part, use ln |u|.

4. On the second, complete the square and use a trigonometric substitution. It should always be a tangent substitution. (If it is a sine or secant substitution, then you could have factored the quadratic.)