Theory

Theory 1

A rational function is a ratio of polynomials, for example:

Partial fraction decomposition

The partial fraction decomposition of a rational function is a way of writing it as a sum of simple terms, like this:

Allowed denominators:

• Linear, e.g. , or linear power, e.g.
• Quadratic, e.g. , or quadratic power, e.g.
  • Condition: quadratics must be irreducible. (No roots, i.e. .)

Allowed numerators: constant (over linear power) or linear (over quadratic power)

These are allowed as simple terms in partial fraction decompositions:

These are not allowed:

These are allowed, showing irreducible quadratic and higher powers:

In this example the numerator is linear and the denominator is quadratic and irreducible.

To create a partial fraction decomposition, follow these steps:

1. Check denominator degree is higher
   • Else do long division
2. Factor denominator completely (even using irrational roots)
3. Write the generic sum of partial fraction terms with their constants

Repeated factors – special treatment – incrementing powers

4. Solve for constants

Theory 2

Partial fractions can be integrated using just a few techniques. Consider these terms:

Linear power bottom

In order to integrate terms like this:

If then use log:

If then use power rule:

Quadratic bottom, constant top

Formula for simple irreducible quadratics:

Memorize this formula!

Quadratic bottom, linear top

In order to integrate terms like this:

Break into separate terms:

Then:

• First term with in top:

• Second term lacking in top:

Extra - Completing the square when “no real roots”

To integrate terms with more general quadratics, like this:

we need , i.e. “no real roots” of the quadratic. If that holds, then we can complete the square and substitute as follows.

Look what happens when completing the square:

Notice that is equivalent to the condition . Create a new label . So this condition means and we can safely define .

Then a -substitution simplifies the equation like this:

The quadratic formula shows that the condition is equivalent to the condition “no real roots.” (In our case . If we had , we could divide out this and change the others.)

So we see that “no real roots” is equivalent to the condition that the denominator can be converted to the format with some constant .

At this point, to compute the integral, do trig sub with and :

Extra - “Rationalize a quotient” - convert into PFD

Sometimes an integrand may be converted to a rational function using a substitution.

Consider this integral:

Set , so and :

Now this rational function has a partial fraction decomposition:

It is easy to integrate from there!

Practice exercises:

• To compute , try the substitution .
• To compute , try the substitution .
• To compute , try the substitution .