7: Partial fractions

7: Partial Fractions

Partial fraction decomposition is a powerful algebraic technique for integrating rational functions (ratios of polynomials). Its core idea is breaking down a complex fraction into a sum of simpler fractions whose integrals are known or easier to find. This method is indispensable when dealing with rational functions where the denominator can be factored.

The Process:

1. Ensure Proper Fraction: Verify the degree of the numerator is less than the degree of the denominator. If not, perform polynomial division first to obtain a polynomial plus a proper rational function. Focus decomposition on the proper fraction part.
2. Factor Denominator: Completely factor the denominator into irreducible factors. These are typically:
   • Linear factors: $(ax + b)$
   • Repeated linear factors: $(ax + b)^k$
   • Irreducible quadratic factors: $(ax^2 + bx + c)$ (cannot be factored into real linear factors, i.e., discriminant $b^2 - 4ac < 0$)
   • Repeated irreducible quadratic factors: $(ax^2 + bx + c)^m$
3. Set Up Decomposition: Write the fraction as a sum of partial fractions based on the denominator factors:
   • For each distinct linear factor $(ax + b)$, include a term: $\frac{A}{ax + b}$
   • For each repeated linear factor $(ax + b)^k$, include terms: $\frac{A_1}{ax + b} + \frac{A_2}{(ax + b)^2} + \dots + \frac{A_k}{(ax + b)^k}$
   • For each distinct irreducible quadratic factor $(ax^2 + bx + c)$, include a term: $\frac{Bx + C}{ax^2 + bx + c}$
   • For each repeated irreducible quadratic factor $(ax^2 + bx + c)^m$, include terms: $\frac{B_1x + C_1}{ax^2 + bx + c} + \frac{B_2x + C_2}{(ax^2 + bx + c)^2} + \dots + \frac{B_mx + C_m}{(ax^2 + bx + c)^m}$
     $A, B, C$, etc., are constants to be determined.
4. Solve for Constants: Multiply both sides of the equation by the original denominator to clear fractions. This yields a polynomial equation true for all $x$. Solve for the unknown constants ($A, B, C$, etc.) using one of two main methods:
   • Substitution: Substitute strategically chosen values of $x$ (often the roots of the denominator factors) to create simpler equations.
   • Equating Coefficients: Expand the polynomial equation and equate the coefficients of corresponding powers of $x$ on both sides. Solve the resulting system of linear equations.
5. Integrate: Once the decomposition is found, integrate each partial fraction term separately. Key results used are:
   • $\int \frac{A}{ax + b} dx = \frac{A}{a} \ln|ax + b| + C$
   • $\int \frac{A}{(ax + b)^k} dx = \frac{A}{a(1-k)} \cdot \frac{1}{(ax + b)^{k-1}} + C$ (for $k > 1$)
   • $\int \frac{Bx + C}{ax^2 + bx + c} dx$ often requires completing the square in the denominator and then splitting into an $\ln$ integral and an $\arctan$ integral.

Example:

Decompose $\frac{x + 5}{x^2 + x - 2}$.

1. Factor denominator: $x^2 + x - 2 = (x + 2)(x - 1)$.
2. Set up: $\frac{x + 5}{((x + 2)(x - 1))} = \frac{A}{x + 2} + \frac{B}{x - 1}$.
3. Multiply by $(x + 2)(x - 1)$: $x + 5 = A(x - 1) + B(x + 2)$.
4. Solve for $A$ and $B$:
   • Substitute $x = 1$: $1 + 5 = A(1 - 1) + B(1 + 2)$ $\Rightarrow$ $6 = 3B$ $\Rightarrow$ $B = 2$.
   • Substitute $x = -2$: $-2 + 5 = A(-2 - 1) + B(-2 + 2)$ $\Rightarrow$ $3 = -3A$ $\Rightarrow$ $A = -1$.
5. Result: $\frac{x + 5}{x^2 + x - 2} = -\frac{1}{x + 2} + \frac{2}{x - 1}$.
6. Integrate: $\int \left[ -\frac{1}{x + 2} + \frac{2}{x - 1} \right] dx = -\ln|x + 2| + 2\ln|x - 1| + C = \ln\left| \frac{(x - 1)^2}{|x + 2|} \right| + C$.

Mastering partial fractions hinges on accurate factoring and systematic solving for constants. Practice decomposing fractions with distinct roots, repeated roots, and quadratic factors.