Partial Fractions

Understanding Partial Fractions

Partial fractions are the expression of a fraction as the sum or difference of simpler fractions.
Any proper rational function (where the degree of the numerator is less than the degree of the denominator) can be expressed as a sum of simpler fractions.
The process of converting a rational function into an equivalent sum of simpler fractions is called partial fraction decomposition.

For simple linear factors in the denominator, the corresponding partial fraction has the form A/(ax+b).
For repeated linear factors, each factor corresponds to a separate term in the partial fraction decomposition, with powers ranging from 1 up to the multiplicity of the root.
When the denominator has irreducible quadratic factors, the corresponding partial fraction has the form (Ax+B)/(ax^2+bx+c).

Start by factoring the denominator of the rational function fully. This identifies the individual fractions to which the rational function will be decomposed.
Each factor, or each different factor raised to any power, contributes to a term in the partial fraction decomposition.
Equate the rational function to the sum of its partial fractions, and collect like terms.
Equating coefficients for the corresponding powers of x gives a system of linear equations in the coefficients of the partial fractions.
Solve the system of linear equations to find the coefficients of the partial fractions.

When the denominator has a quadratic factor that can’t be easily factored, use completing the square to turn it into a form where you can recognise a standard integral to be able to integrate.

Integration and differential equations often become much simpler when the integrand is expressed as partial fractions rather than as a single, more complex fraction.
Notably, partial fractions are used in the integration of rational functions and in finding inverse Laplace transforms.

It’s crucial to remember that a partial fraction decomposition always exists for a proper rational function. However, such a decomposition may not exist if the degree of the numerator is greater than or equal to the degree of the denominator. This is a common mistake to avoid.