--- title: "Partial Fractions for Rational Integrands" description: "Check degrees, divide when needed, and split a rational integrand into fractions that integrate to logarithmic, rational or arctangent terms." canonical: https://lightmysky.com/learn/mathematics/partial-fractions-for-rational-integrands-mt_PA8qxs424c source: https://lightmysky.com/learn/mathematics/partial-fractions-for-rational-integrands-mt_PA8qxs424c.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Partial Fractions for Rational Integrands Check degrees, divide when needed, and split a rational integrand into fractions that integrate to logarithmic, rational or arctangent terms. Subject: Mathematics · Area: Calculus & Analysis · Ages 19 to 20 Page: https://lightmysky.com/learn/mathematics/partial-fractions-for-rational-integrands-mt_PA8qxs424c ## Ready when they can - Divide first when the numerator's degree is at least the denominator's, and identify the polynomial quotient and proper remainder - Set up full partial-fraction templates for repeated linear factors and irreducible quadratic factors - Respect excluded inputs, solve and check coefficients, integrate the pieces, combine logarithms, and evaluate definite integrals on intervals without denominator zeros ## Lesson: Split fractions before you integrate Compare the numerator's and denominator's degrees. If the numerator's degree is at least the denominator's, divide first. Divide leading terms, multiply by the divisor, subtract, and repeat until the remainder has lower degree. For example, (x^2+6x+4)/(x+2)=x+4-4/(x+2). Integrate the polynomial quotient separately. Apply partial fractions to the proper remainder, whose numerator has lower degree than its denominator. Factor the proper denominator. Give each distinct linear factor a constant numerator, then multiply by the full denominator and equate numerators. Solve by substitution or by matching coefficients of like powers. The resulting polynomial identity holds for all real x, so denominator zeros may be substituted there. They remain excluded from the original fractions. Check the split by recombining over a common denominator. **Example.** Split 6/((x-2)(x+4)) as A/(x-2)+B/(x+4). Clearing denominators gives 6=A(x+4)+B(x-2). In this polynomial identity, x=2 gives 6=6A, so A=1. At x=-4, 6=-6B, so B=-1. Thus the split is 1/(x-2)-1/(x+4). Its antiderivative is ln|x-2|-ln|x+4|+C on each interval avoiding -4 and 2. For a repeated linear factor (ax+b)^n, include a constant numerator over every power from 1 through n in the full template. Add terms for other factors too. Thus x^2(x+3) needs A/x+B/x^2+C/(x+3). Some coefficients may turn out to be zero. For (px+q)/(x+r)^2, clearing denominators gives px+q=A(x+r)+B=Ax+Ar+B. Match x coefficients and constants: A=p and B=q-pr. *(drawing: For (4x+9)/(x+2)^2, the split is 4/(x+2)+1/(x+2)^2. Match both the x term and the constant.)* For a≠0, ∫A/(ax+b) dx=(A/a) ln|ax+b|+C. For an integer n>1, ∫B/(ax+b)^n dx=B(ax+b)^(1-n)/(a(1-n))+C. A first-power linear denominator gives a logarithm. Higher powers give rational terms. For example, ∫3/(x-5)^2 dx=-3/(x-5)+C. Differentiate to check the sign, power and scale factor. Work on intervals where the denominator is nonzero. An irreducible quadratic is a quadratic with no real roots. Its numerator must allow a linear expression. For x(x^2+a^2), with a>0, use A/x+(Bx+D)/(x^2+a^2). If a quadratic factor is repeated, put a separate linear numerator over every required power. A constant numerator alone can miss the x term needed in the general template. For a>0, ∫(Mx+N)/(x^2+a^2) dx=(M/2) ln(x^2+a^2)+(N/a) arctan(x/a)+C. Split the numerator: substitution u=x^2+a^2 handles Mx, while the constant term uses the arctangent formula. Arctan means inverse tangent, measured in radians. For example, (2x+3)/(x^2+9) integrates to ln(x^2+9)+arctan(x/3)+C. A quadratic piece can contribute both functions. Combine logarithms using their power and quotient rules. For nonzero u and v, 2 ln|u|-ln|v|=ln|u^2/v|. The coefficient becomes an exponent, and subtraction becomes division inside the logarithm. Combining an answer does not restore any input excluded by the original denominator. A sum of logarithms is also a valid answer; combining them is optional. For a definite integral, first check the interval for denominator zeros. If the integrand is continuous throughout [a,b] and F is an antiderivative there, evaluate F(b)-F(a). Do not apply this ordinary rule across a denominator zero without a separate check of limits. For the fraction 6/((x-2)(x+4)), F(x)=ln|(x-2)/(x+4)|. From 4 to 8, subtract ln(1/4) from ln(1/2) to get ln 2. Keep logs exact, or round if asked. **Recap.** Divide if needed, choose the full template, solve and check coefficients, then integrate. First-power linear terms give logs; higher powers give rational terms; quadratic pieces can give arctangents. *The lesson, practice questions, worked solutions and flashcards were written for this dataset.* ## Practice 17 questions on this page, each with its working shown. ## Optional connections - [Polynomial Division and the Factor Theorem](https://lightmysky.com/learn/mathematics/polynomial-division-and-the-factor-theorem-mt_haL2UkGa5C) ## Opens up - [The Laplace Transform](https://lightmysky.com/learn/mathematics/the-laplace-transform-mt_8j8GXzITN0) - [Improper Integrals and Their Convergence](https://lightmysky.com/learn/mathematics/improper-integrals-and-their-convergence-mt_gvtZPFRA6Z)