Partial Fractions

Level 1 - Math I (Physics) topic page in Integration.

Partial Fraction Decomposition

Partial fractions decompose rational functions into simpler fractions that are easier to integrate.

For proper rational functions \(\frac{P(x)}{Q(x)}\) where \(\deg(P) < \deg(Q)\). If \(\deg(P) \geq \deg(Q)\), first perform polynomial division.

For \(\frac{P(x)}{(x-a)(x-b)(x-c)}\):

Linear Distinct

\[\frac{P(x)}{(x-a)(x-b)(x-c)} = \frac{A}{x-a} + \frac{B}{x-b} + \frac{C}{x-c}\]

For \(\frac{P(x)}{(x-a)^2(x-b)}\):

Linear Repeated

\[\frac{P(x)}{(x-a)^2(x-b)} = \frac{A}{x-a} + \frac{B}{(x-a)^2} + \frac{C}{x-b}\]

For \(\frac{P(x)}{(x^2+ax+b)(x-c)}\):

Quadratic

\[\frac{P(x)}{(x^2+ax+b)(x-c)} = \frac{Ax+B}{x^2+ax+b} + \frac{C}{x-c}\]

Multiply both sides by the denominator and equate coefficients, or use the cover-up method for linear factors.

Decompose \(\frac{3x+5}{(x-1)(x+2)}\):

Setup

\[\frac{3x+5}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}\]

Multiply: \(3x+5 = A(x+2) + B(x-1)\)

Coefficients

\[3x+5 = (A+B)x + (2A-B)\]

Equate: \(A+B=3\) and \(2A-B=5\)

Solving: \(A=2\), \(B=1\)

Result

\[\frac{3x+5}{(x-1)(x+2)} = \frac{2}{x-1} + \frac{1}{x+2}\]

After decomposition:

Integration

\[\int \frac{2}{x-1} \, dx + \int \frac{1}{x+2} \, dx = 2\ln|x-1| + \ln|x+2| + C\]