AcademyIntegral Calculus
Academy
Partial Fractions
Level 1 - Calculus topic page in Integral Calculus.
Principle
Partial fractions rewrite a rational function as a sum of simpler rational functions. This makes integration possible using logarithms and basic antiderivatives.
Notation
\(\frac{P(x)}{Q(x)}\)
rational function
\(A,B\)
constants to determine
\(Q(x)\)
denominator polynomial
The Core Method
Factor the denominator, write the correct partial fraction form, multiply through by the denominator, and solve for constants.
Linear factors
\[\frac{P(x)}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b}\]
If the numerator degree is not smaller than the denominator degree, divide first.
Worked Cases
Question
Find \(\int \frac{3x+5}{(x+1)(x+2)}\,dx\).
Answer
Write \(\frac{3x+5}{(x+1)(x+2)}=\frac{A}{x+1}+\frac{B}{x+2}\). Multiply through: \(3x+5=A(x+2)+B(x+1)\). Set \(x=-1\): \(2=A\). Set \(x=-2\): \(-1=-B\), so \(B=1\). The integral is \(2\ln|x+1|+\ln|x+2|+C\).
Examples
Question
What form is used for \(\frac{1}{x(x-1)^2}\)?
Answer
Use one term for \(x\), and one term for each power of the repeated factor: \(\frac{1}{x(x-1)^2}=\frac{A}{x}+\frac{B}{x-1}+\frac{C}{(x-1)^2}\).