# Partial Fractions ![[Readwise/Articles/Partial Fractions#^tf9v88]] Partial fraction decomposition works whenever the denominator expression can be fully factored out into linear expressions and irreducible quadratic expressions. ^ill539 All of the techniques described below assume that the degree of denominator is greater than the degree of the numerator. If that is not the case, long division should be applied first. ## Partial Fraction Decomposition With Linear Factors When the denominator of rational function can be factored into a linear binomials the following should be used to complete the composition. For example ![[Partial Fractions - Linear Factors#^ex7sa9]] needs to be broken down into ![[Partial Fractions - Linear Factors#^xbzp59]] This then yields a system of equations with two unknowns that can be solved for $A$ and $B$, ultimately yielding ![[Partial Fractions - Linear Factors#^1sh1ze]] The number of partial fractions, and therefore the number of equations that need to be solved (and the number of unknowns) will always correspond to the degree of the denominator. Instead of solving a system of equations the cover up rule can be used instead (see below). For the full example see [[Partial Fractions - Linear Factors]]. Integrate any of these partial fractions by pulling out the coefficient and applying $\int \frac 1 {x \pm a} \, dx = \ln |x\pm a| + C$ ## Partial Fraction Decomposition With Repeated Factors ![[Partial Fractions - Repeated Factors#^0nqmdq]] For example, when doing decomposition of the following fraction ![[Partial Fractions - Repeated Factors#^wvdtf7]] because ![[Partial Fractions - Repeated Factors#^4ikfph]] you would decompose the fraction like: ![[Partial Fractions - Repeated Factors#^iiara5]] The steps for finding $A$, $B$, and $C$ are then the same as for linear factors. For the full example see [[Partial Fractions - Repeated Factors]]. Repeated linear terms can be integrated by first substituting the linear expression and then: $\int \frac 1 {u^b} \, du = \int u^{-b} \, du = \frac 1 {-b+1} u^{-b + 1} + C= \frac {u^{-b+1}} {-b + 1} + C$ (for $b \ne 1$) ## Partial Fraction Decomposition With Irreducible Quadratics ![[Partial Fractions - Irreducible Quadratics#^y2w85n]] Example: ![[Partial Fractions - Irreducible Quadratics#^d0oajk]] The coefficients should be determined the same way. For a full example see [[Partial Fractions - Irreducible Quadratics]]. Irreducible quadratic partial fractions can be integrated by first splitting them into a form where the numerator is the derivative of the denominator (by polynomial long division) plus some term that will have 1 in the numerator. The purpose of this is that the integral of a fraction where the numerator is the derivative of the denominator is known: ![[Antiderivative#^idur7o]] Example: $ \begin{align} \int \frac {5x+3}{x^2 -x +1} \, dx &= \int \frac {\frac 5 2 (2x-1) + \frac {11} 2} {x^2 -x +1} \, dx \\ &=\frac 5 2 \int \frac {2x-1} {x^2 -x +1} \, dx + \frac {11} 2 \int \frac 1 {x^2 -x +1} \, dx \\ &= \frac 5 2 \ln |x^2-x+1| + \frac {11} 2 \int \frac 1 {x^2 -x +1} \, dx \end{align} $ The irreducible quadratics with 1 in the numerator can be integrated like: ![[Antiderivatives of Reciprocal Quadratics#^33xemi]] ![[Antiderivatives of Reciprocal Quadratics#^vkb1ui]] ## The Cover Up Rule The cover up rule is a technique that makes it simpler to calculate the individual numerators of partial fractions (the coefficients $A$, $B$, $C$,...). For example, if we're trying to decompose a rational fraction whose denominator is of third degree we'd write: ![[Partial Fractions - Cover Up Rule#^tb68vv]] If we then rationalize this we'll get: $f(x) = A(x-b)(x-c) + B(x-a)(x-c) + C(x-a)(x-b)$ This equality is valid for values of $x$, meaning that if we pick $x=a$ the second and third terms will be eliminated allowing us to easily compute $A$. The process is then repeated for $B$ and $C$, leading to the following formulas: ![[Partial Fractions - Cover Up Rule#^63b4hh]] ![[Partial Fractions - Cover Up Rule#^ymbuaa]] The proces is similar for higher degrees. For a full example see [[Partial Fractions - Cover Up Rule]]. ## The Limit Method for Determining Coefficients ![[Partial Fractions - Limit Method#^6fifss]] For example look at ![[Partial Fractions - Limit Method#^yd22hj]] ![[Partial Fractions - Limit Method#^l9q2gi]] ![[Partial Fractions - Limit Method#^9g7ukv]] For a full example look at [[Partial Fractions - Limit Method]].