Partial Fraction Decomposition

Breaking Rational Functions for Integration

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Partial fraction decomposition is a method for rewriting a complicated rational expression as a sum of simpler fractions. It matters in calculus because many integrals are much easier to evaluate after this rewrite. The idea is especially useful when the denominator factors into linear or irreducible quadratic pieces. It also helps in differential equations, Laplace transforms, and algebraic simplification.

The method starts by factoring the denominator completely and then choosing a fraction form for each factor. Unknown constants are placed in the numerators, and those constants are found by clearing denominators and matching coefficients or substituting convenient values. For distinct linear factors, the numerators are constants such as A or B. For repeated factors and quadratic factors, the setup changes in a predictable way, so choosing the correct form is a key step.

Key Facts

For distinct linear factors, $\frac{P(x)}{(x - a)(x - b)} = \frac{A}{x - a} + \frac{B}{x - b}$ , provided degree of $P$ is less than degree of denominator.
Example: $\frac{3x + 1}{(x - 1)(x + 2)} = \frac{A}{x - 1} + \frac{B}{x + 2}$ .
Clear denominators by multiplying both sides by the full denominator, then solve for constants.
Coefficient matching means expanding and comparing powers of x on both sides.
For a repeated linear factor, use separate terms: $\frac{P(x)}{(x - a)^2} = \frac{A}{x - a} + \frac{B}{(x - a)^2}$ .
For an irreducible quadratic factor, use a linear numerator: $\frac{P(x)}{x^2 + 1} = \frac{Ax + B}{x^2 + 1}$ .

Vocabulary

Rational function: A rational function is a quotient of two polynomials, such as P(x)/Q(x).
Partial fraction decomposition: Partial fraction decomposition is the process of rewriting a rational function as a sum of simpler fractions.
Linear factor: A linear factor is a first degree factor of the form $x - a$ or $x + b$ .
Repeated factor: A repeated factor is a denominator factor that appears more than once, such as $(x - 3)^2$ .
Coefficient matching: Coefficient matching is solving for unknown constants by making the coefficients of like powers of x equal on both sides.

Common Mistakes to Avoid

Using partial fractions before checking degrees, which is wrong because an improper rational function must first be divided so the numerator degree is less than the denominator degree.
Forgetting to factor the denominator completely, which is wrong because the decomposition form depends on the actual factors, not the unfactored polynomial.
Using only one term for a repeated factor, which is wrong because a factor like $(x - a)^2$ requires both $\frac{A}{x - a}$ and $\frac{B}{(x - a)^2}$ .
Putting a constant numerator over an irreducible quadratic, which is wrong because a factor like $x^2 + 1$ needs a linear numerator $Ax + B$ .

Practice Questions

1 Decompose (7x + 1)/[(x - 2)(x + 3)] into partial fractions and solve for the constants.
2 Decompose $\frac{2x^2 + 3x + 4}{(x - 1)^2}$ into partial fractions. Write the correct form first, then find the constants.
3 Explain why the decomposition of $\frac{5}{(x - 1)(x^2 + 4)}$ must use the form $\frac{A}{x - 1} + \frac{Bx + C}{x^2 + 4}$ instead of $\frac{A}{x - 1} + \frac{B}{x^2 + 4}$ .

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