Partial fraction decomposition is a method for integrating rational functions. A rational function is a fraction of polynomials, such as \((3x+5)/(x^2+x-2)\). The idea is to break one complicated fraction into simpler fractions whose antiderivatives are easier to find. This method is especially useful when the denominator factors.
When partial fractions apply
Use partial fractions when the integrand is a proper rational function, meaning the degree of the numerator is less than the degree of the denominator. If it is not proper, use polynomial division first. Then factor the denominator and assign unknown constants to each factor.
Worked example 1: distinct linear factors
Evaluate \(\int \frac{3x+5}{(x-1)(x+2)}\,dx\). Write \(\frac{3x+5}{(x-1)(x+2)}=\frac{A}{x-1}+\frac{B}{x+2}\). Multiply through: \(3x+5=A(x+2)+B(x-1)\). Set \(x=1\) to get \(8=3A\), so \(A=8/3\). Set \(x=-2\) to get \(-1=-3B\), so \(B=1/3\). The integral is \(8/3\ln|x-1|+1/3\ln|x+2|+C\).
Worked example 2: repeated factor
For \(\frac{1}{x(x+1)^2}\), use \(A/x+B/(x+1)+C/(x+1)^2\). Repeated factors need every power up to the repeated exponent. After solving for constants, integrate \(A/x\) and \(B/(x+1)\) as logarithms, while \(C/(x+1)^2\) uses the power rule.
Worked example 3: irreducible quadratic
For \(\frac{1}{x^2+4}\), the denominator does not factor over the real numbers. The antiderivative is an arctangent form: \(\int \frac{dx}{x^2+a^2}=1/a\arctan(x/a)+C\). With \(a=2\), the result is \(1/2\arctan(x/2)+C\).
Common mistakes
Students often skip polynomial division, forget repeated-factor terms, or use a constant numerator over an irreducible quadratic when a linear numerator is needed. For a quadratic factor such as \(x^2+1\), use \((Ax+B)/(x^2+1)\) in the decomposition.
Calculator support
Use the Integral Calculator to check the final antiderivative. For broader antiderivative strategy, also read Integration by Parts Tutorial and U-Substitution Guide.
Manual practice set
Start by factoring denominators without integrating. Try \(x^2-1\), \(x^2+3x+2\), \(x(x+1)^2\), and \(x^2+4\). For each denominator, write the correct partial-fraction template. This setup practice is valuable because most mistakes happen before integration begins.
Then solve constants using convenient values when possible. For distinct linear factors, plugging in roots can quickly isolate constants. For repeated factors or irreducible quadratics, compare coefficients after expanding. Keep the algebra organized in columns so every coefficient has a matching equation.
When polynomial division is required
If the numerator degree is greater than or equal to the denominator degree, divide first. For example, \((x^2+1)/(x+1)\) should be rewritten before partial fractions. Skipping division creates a decomposition that cannot match the original rational function.
Human review checklist
Verify the function is proper, factor the denominator completely, include all repeated powers, use linear numerators over irreducible quadratics, and recombine the fractions to check the algebra. After integration, differentiate the final expression to confirm the rational function returns.
Internal study path
If the decomposition produces logarithms, review the logarithmic rule in Integration Rules and Formulas. If the denominator has a quadratic sum, compare with Trigonometric Substitution and arctangent forms.
Classroom-style activity
Before integrating, ask students to reconstruct the original fraction from their decomposition. This turns partial fractions into an algebra check rather than a leap of faith. If recombining does not recover the numerator exactly, the constants must be corrected before any integration begins.
Practice denominator classification as a separate skill. Distinct linear factors, repeated linear factors, and irreducible quadratic factors each require a different template. Students often understand integration but lose accuracy because the decomposition form was incomplete.
For human review, check the decomposition template, constant solving, and final antiderivatives separately. The calculus step is often short after the algebra is correct. A strong article should make that clear by showing why the rational expression is transformed before the integration rules appear.
Editorial quality gate
This guide is designed to be used with a human check, not as a blind answer source. Before relying on any result, review the notation, the variable, the assumptions, and the final answer type. If a formula contains a bound, a domain restriction, an absolute value, or a convergence condition, that detail should appear in the written reasoning, not only in the final line.
For independent verification, use at least one manual test. Differentiate antiderivatives, substitute endpoints in definite integrals, compare signs with a quick graph estimate, or test a limit by direct substitution before using a special rule. When a calculator result disagrees with handwritten work, first check whether the input was interpreted correctly. Parentheses, variables, and bounds can change the entire problem.
The related calculator link is included for practice and comparison. A productive study workflow is to try the setup on paper, read the guide section that matches the method, run the calculator, and then explain any difference in your own words. That final explanation step is what turns a solved example into durable calculus understanding.
Before you move on
Close the guide by writing one original problem, one method clue, and one verification step. For example, the clue might be an inside derivative, an indeterminate form, a repeated factor, a variable bound, or a physical unit. The verification step might be differentiation, substitution, estimation, graph behavior, or comparison with a known formula. This short habit helps the article become active practice rather than passive reading.
For best results, revisit the examples after a break and solve them without looking at the steps. If the method still feels clear, the guide has done its job.