Integration by Partial Fractions

Integration by Partial Fractions (a.k.a. Integration by Decomposition) is a method used to decompose and then integrate a rational fraction integrand that has complex terms in the denominator. By using partial fraction, we calculate and decompose the expression into simpler terms so that we can easily calculate or integrate the expression thus obtained. The basic idea in the integration by partial fractions is to factor the denominator and then decompose them into two different fractions where the denominators are the factors respectively and the numerator is calculated suitably.

Contents

Integration of Partial Fractions Technique

The integration technique of partial fractions is a way to integrate rational functions (Recall that a rational function is one polynomial divided by another.) of the form

$f\left( x \right) =\frac{P\left( x \right)}{Q\left( x \right)}$

What is a Partial Fraction?

Partial fractions are the simpler, individual fractions that are added or subtracted together to create a single, more complex fraction. The process of working backwards—breaking a single complex fraction down into these smaller, constituent parts—is called partial fraction decomposition. It essentially reverses the process of finding a common denominator to add fractions together

(Recall that f is a rational function when P(x) and Q(x) are polynomials.) When considering

$\int{\frac{P\left( x \right)}{Q\left( x \right)}dx}$

first look for a simple substitution, as with any integral. If you see a way to use integration by parts, or even trigonometric substitution, you should probably try this first, as those methods can be a little simpler. Sometimes partial fraction decomposition (“unadding”) is the obvious and only choice.

The partial fractions method can be used to integrate rational functions The basic idea behind the partial fraction approach is “unadding” a fraction:

$\frac{5}{6}=\frac{1}{2}+\frac{1}{3}$

Before using the partial fractions technique, you have to check that your integrand is a “proper” fraction — that’s one where the degree of the numerator is less than the degree of the denominator. If the integrand is “improper,” like

$\int{\frac{2x^3+x^2-10}{x^3-3x-2}}$

you have to first do long polynomial division to transform the improper fraction into a sum of a polynomial (which sometimes will just be a number) and a proper fraction.

How to Integrate using Partial Fractions

To evaluate the integral ∫ p(x)/q(x) dx where p(x)/q(x) is in a proper rational fraction, we can factorize the denominator, then using the following rational fraction cases we can write the integrand in a form of the sum of simpler rational functions including constant A, B, C, etc. Then values of A, B, C, etc. can be obtained using various methods of algebra.

Calculus Fact

The key in the partial fractions technique of integration is to decompose P(x)/Q(x) into a sum of simpler fractions, whose denominators are related to the factors of Q(x).

Once we’ve determined that partial fractions can be done we factor the denominator as completely as possible. Then for each factor in the denominator we can use the following table to determine the term(s) we pick up in the partial fraction decomposition.

*Partial Fraction Formula – Paul’s Online Notes*

Notice that the first and third cases are really special cases of the second and fourth cases respectively.

To integrate any rational function using Partial Fractions, we need to follow the following steps:

Step 1: Factor the denominator given rational function into linear and quadratic factors.
Step 2: Use the Partial Fraction Formula to write the rational function as a sum of simpler fractions.
Step 3: Determine the constants A, B, and C.
Step 4: Integrate each partial fraction separately with appropriate methods to get the final integral.

See the solution to Example 1 on how to evaluate the following integral.

$\int{\frac{3x+11}{x^2-x-6}dx}$

Summary of Integration by Partial Fractions

Rational Functions: A rational function is the ratio of two polynomials, with the denominator not equal to zero.
Proper Rational Functions: The degree of the numerator is less than the denominator, allowing partial fraction decomposition.
Improper Rational Functions: The degree of the numerator is greater than or equal to the denominator. Polynomial division is used to separate the function into a polynomial quotient and a proper rational remainder.
Partial Fraction Decomposition: A method to break a rational function into simpler fractions, depending on the form of the denominator (linear, repeated linear, or irreducible quadratic factors).
- Case I: Non-repeating linear factors in the denominator lead to a sum of fractions, with constants determined by substitution or coefficient comparison. (Non-repeating linear factors are algebraic terms of the form (ax + b) that appear only once in the denominator of a rational expression.)
- Case II: Repeating linear factors require decomposition into terms for each power, with constants found by substitution or differentiation. (Repeating linear factors are algebraic terms of the form (ax + b)ⁿ, where n > 1, that appears in the denominator of a rational expression.)
- Case III: Non-repeating irreducible quadratic factors require linear numerators in the decomposition, with constants found by expanding and comparing coefficients.
- Case IV: Repeating irreducible quadratic factors require terms for each power, with constants found through substitution and coefficient comparison. [Khandelwal]

References

[ ℰ ] Dawkins, Paul. “Calculus II.” Section 7.4 : Partial Fractions. Paul’s Online Notes, May 11, 2026. https://tutorial.math.lamar.edu/classes/calcii/partialfractions.aspx.

“How to Integrate by Using Partial Fractions When the Denominator Contains Only Linear Factors.” Dummies, March 26, 2016. https://www.dummies.com/article/academics-the-arts/math/calculus/how-to-integrate-by-using-partial-fractions-when-the-denominator-contains-only-linear-factors-192159/.

“Integration by Partial Fractions.” GeeksforGeeks, January 1, 2021. https://www.geeksforgeeks.org/maths/integration-by-partial-fractions/.

“Integration by Partial Fractions – Definition, Formula, Examples.” CUEMATH, Accessed July 19, 2026. https://www.cuemath.com/calculus/integration-by-partial-fractions/.

[ ℰ ] Khandelwal, Neha. “Integration by Partial Fractions.” CK-12 Foundation, July 2, 2025. https://flexbooks.ck12.org/cbook/ck-12-cbse-maths-class-12/section/7.5/primary/lesson/integration-by-partial-fractions/.

Org, Dash Hrecos. “Integrals Using Partial Fractions.” Dash Hrecos Org, July 8, 2026. https://www.dash.hrecos.org/explore/178/5AD/o1dTVb/IntegralsUsingPartialFractions.

“Strategies of Integration.” web.ma.utexas.edu, Accessed July 19, 2026. https://web.ma.utexas.edu/users/m408s/CurrentWeb/LM7-5-6.php.

Additional Reading

“Integration by Partial Fractions.” 8.4 Integration by Partial Fractions. CK-12 Foundation, November 29, 2023. https://flexbooks.ck12.org/cbook/ck-12-calculus-concepts/section/8.4/primary/lesson/integration-by-partial-fractions-calc/.

OpenStax. “11.4: Partial Fractions.” LibreTexts, October 27, 2016. https://math.libretexts.org/Bookshelves/Algebra/Algebra_and_Trigonometry_1e_(OpenStax)/11%3A_Systems_of_Equations_and_Inequalities/11.04%3A_Partial_Fractions.

“Partial Fractions.” Brilliant, Accessed July 19, 2026. https://brilliant.org/wiki/partial-fractions/.

“Partial Fractions.” CK-12 Foundation, November 27, 2023. https://flexbooks.ck12.org/cbook/ck-12-precalculus-concepts-2.0/section/8.10/related/lesson/integration-by-partial-fractions-calc/.

“Partial Fraction Decomposition.” danville.edu, Accessed July 19, 2026. https://danville.edu/sites/default/files/assets/files/Math%20Lab/Partial%20Fraction%20Decomposition.pdf.

Pierce, Rod. “Partial Fractions.” Math Is Fun. Accessed July 19, 2026. https://www.mathsisfun.com/algebra/partial-fractions.html.

Spong, Mark. “Partial Fractions.” CK-12 Foundation, November 27, 2023. https://flexbooks.ck12.org/cbook/ck-12-precalculus-concepts-2.0/section/8.10/primary/lesson/partial-fractions-pcalc/.

Stapel, Elizabeth. “What’s a Partial Fraction? How Does It Decompose?” Purplemath. Accessed July 19, 2026. https://www.purplemath.com/modules/partfrac.htm.

Stapel, Elizabeth. “How to Work with Repeated or Irreducible Factors.” Purplemath. Accessed July 19, 2026. https://www.purplemath.com/modules/partfrac2.htm.

Strang, Gilbert, and Edwin “Jed” Herman. “7.4: Partial Fractions.” LibreTexts, July 11, 2016. https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/07%3A_Techniques_of_Integration/7.04%3A_Partial_Fractions.

Practice Problems

Dawkins, Paul. “Calculus II.” Partial Fractions (Practice Problems), November 16, 2022. https://tutorial.math.lamar.edu/problems/calcii/partialfractions.aspx.

Kouba, Duane. “Integration by Partial Fractions.” math.ucdavis.edu, May 1, 2000. https://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/partialfracdirectory/PartialFrac.html.

OpenStax. “7.4E: Exercises for Integration by Partial Fractions.” LibreTexts, March 12, 2019. https://math.libretexts.org/Courses/Monroe_Community_College/MTH_211_Calculus_II/Chapter_7%3A_Techniques_of_Integration/7.4%3A_Partial_Fractions/7.4E%3A_Exercises_for_Integration_by_Partial_Fractions.

Strang, Gilbert, and Edwin “Jed” Herman. “7.4E: Exercises for Section 7.4.” LibreTexts, June 23, 2021. https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/07%3A_Techniques_of_Integration/7.04%3A_Partial_Fractions/7.4E%3A_Exercises_for_Section_7.4.

Notes

What is a repeated factor?

A repeated factor is a factor which is raised to a power — like (x − 3)⁴ — or otherwise occurs in a rational expression’s denominator more than once.

What is an irreducible factor?

An irreducible factor is a quadratic factor which does not itself factor into two linear polynomials. If plugging the quadratic into the Quadratic Formula generates answers with square roots or complex values, then (in the context of partial fraction decomposition) the quadratic is irreducible. [Purplemath]

Irreducible factors are the rational-expression version of prime-number factors in regular fractions.

If the denominator of your rational expression has an unfactorable quadratic, then you have to account for the possible size of the numerator. When the denominators were linear expressions, the numerators were one degree less; tht is, they were constants. If the denominator contains a degree-two factor, then the numerator might not be just a number; it might be of degree one. So you would deal with a quadratic factor in the denominator by including a linear expression in the numerator. [Purplemath]

Videos

Partial Fraction Decomposition (Complete Guide)

Master “partial fraction decomposition” with this complete guide from Mario’s Math Tutoring! This video covers all the different types of partial fraction problems you’ll encounter, showing you how to set them up, solve for the unknown numerators (A, B, C, etc.), and utilize two key methods for solving.

Partial Fraction Decomposition | Math with Professor V

Calculus 2: Integration of Rational Functions by Partial Fractions (Video #4) | Math w/ Professor V

Examples of all four cases involved with using partial fraction decomposition to evaluate integrals. The resulting integrals can involve natural logarithms, inverse trig functions, u-substitution, trigonometric substitution–you name it! An integration potpourri!

Evaluating Integrals Using Partial Fractions/Partial Fraction Decomposition | Math with Professor V

Four examples evaluating indefinite integrals using partial fractions. A continuation to the introductory lesson on this topic.

[ ℰ ] This exceptional reference is highly recommended for your consideration.

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