Partial fractions integration is a method for integrating rational functions, which are fractions made from polynomials. This cheat sheet helps students choose the correct decomposition form before integrating. It is especially useful when a rational expression has factorable denominators, repeated factors, or irreducible quadratic factors.
Students need this reference to avoid guessing and to organize each step clearly.
The main idea is to rewrite a complicated rational function as a sum of simpler fractions. If the numerator degree is greater than or equal to the denominator degree, use polynomial long division first. Linear factors usually lead to logarithms, while irreducible quadratic factors may lead to logarithms and inverse tangent terms.
Correct factor matching is the key to making the integration straightforward.
Key Facts
- A rational function is proper when for , and partial fractions should be applied only after the fraction is proper.
- If , first divide to write , where .
- For a distinct linear factor , use the partial fraction term .
- For a repeated linear factor , use .
- For an irreducible quadratic factor , use the numerator and write .
- For a repeated irreducible quadratic factor , include .
- The basic linear partial fraction integral is .
- A completed square denominator often uses .
Vocabulary
- Rational function
- A rational function is a quotient of polynomials written as , where .
- Proper rational expression
- A proper rational expression has numerator degree less than denominator degree, so .
- Partial fraction decomposition
- Partial fraction decomposition rewrites one rational expression as a sum of simpler rational expressions with known denominator factors.
- Irreducible quadratic
- An irreducible quadratic is a quadratic factor such as that cannot be factored into real linear factors.
- Repeated factor
- A repeated factor is a denominator factor raised to a power greater than , such as or .
- Cover-up method
- The cover-up method is a shortcut for finding constants for distinct linear factors by temporarily removing one factor and substituting its zero.
Common Mistakes to Avoid
- Skipping long division when is wrong because partial fractions require a proper rational expression before decomposition.
- Using only for a repeated linear factor is wrong because every power from to must appear in the decomposition.
- Writing for an irreducible quadratic is wrong because the numerator must be linear, so it should be .
- Forgetting absolute value in is wrong because the antiderivative of is defined using the absolute value of the linear factor.
- Factoring the denominator incorrectly is wrong because one missed or false factor changes every partial fraction term and leads to an invalid integral.
Practice Questions
- 1 Decompose and integrate .
- 2 Evaluate using partial fractions.
- 3 Set up the correct partial fraction form for without solving for the constants.
- 4 Explain why must be handled with polynomial division before partial fraction decomposition.
Understanding Partial Fractions Integration Reference
Factoring the denominator completely over the real numbers is the planning stage. Pull out any common numerical factor first. Then look for differences of squares, grouping patterns, or quadratic factors.
A quadratic that has no real roots must stay as one quadratic factor. The decomposition form is determined by the factors before any coefficients are found. This is why a small factoring error changes the whole answer.
For repeated factors, every power must appear in the setup. Each power represents a separate possible part of the original fraction. Leaving out one term makes it impossible to match the original numerator later.
After writing the full decomposition, multiply every term by the original denominator. This step clears the small denominators and turns the problem into a polynomial identity. The two sides must have equal coefficients for every power of the variable.
Students can find the unknown constants by substituting useful values, especially values that make a factor zero. This shortcut works well for separate linear factors. It does not find every constant when factors repeat or when quadratic factors are present.
In those cases, expand carefully, collect like powers, and solve the resulting system of linear equations. Checking the constants by recombining the fractions is worth the time. It catches sign errors before integration makes them harder to see.
Quadratic terms need extra attention because their numerators can be separated into parts with different integration methods. Compare the numerator with the derivative of the quadratic denominator. One multiple of that derivative produces a natural logarithm after integration.
Any leftover constant may require completing the square. Completing the square rewrites the quadratic as a squared expression plus a positive constant. A substitution then changes the integral into the standard inverse tangent form.
When a quadratic is raised to a power, the same idea may be combined with substitutions or reduction patterns. The key is to avoid treating every quadratic numerator as a constant. A linear numerator is needed because it can represent both the derivative part and the leftover part.
These skills appear whenever a rate is modeled by a ratio of polynomials. Examples include simplified models of motion, electrical circuits, chemical mixing, and population change. In school problems, the main challenge is usually organization rather than advanced calculus.
Write the factorization first, list every required fraction, then solve constants before starting any integrals. Keep absolute value bars with logarithms from linear factors, since the expression inside can be negative on part of its domain.
Finally, differentiate the completed answer when possible. Differentiation is the most reliable check because it should rebuild the original rational function on intervals where that function is defined.