Integrating Using Linear Partial Fractions
Decompose distinct nonrepeating linear factors, verify the algebra, and preserve excluded inputs.
College Board Topic 6.12 · CED p. 124 (PDF p. 129) · CED effective Fall 2020; project course outline 2026–27.
BC extension: a proper rational function with distinct linear factors can split into simpler fractions. Write (3x+5)/[(x+1)(x+2)] = A/(x+1)+B/(x+2). Multiply by the denominator to get 3x+5=A(x+2)+B(x+1). Setting x=−1 gives A=2 and x=−2 gives B=1. Coefficient comparison works too.
Integrate the resulting logarithms: 2ln|x+1|+ln|x+2|+C. If the rational function is improper, first use the shared AB long-division lesson. This required BC topic concerns linear, nonrepeating factors; more elaborate decompositions are enrichment. Check the original denominator throughout the proposed interval before evaluating a definite integral.
Let's work through it
- Decompose 1/[x(x+1)] as A/x+B/(x+1). Reason: The denominator consists of distinct linear factors.
- Solve 1=A(x+1)+Bx: A=1, B=−1. Reason: Multiplying through produces an identity for coefficients.
- Integrate to ln|x|−ln|x+1|+C. Reason: Each fraction has a logarithmic antiderivative on intervals avoiding 0 and −1.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
Why divide first if numerator degree is larger?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Why divide first if numerator degree is larger?
Worked answer and reason
Partial fractions require a proper rational remainder.
Long division separates off the polynomial.
Retrieval 2
Why keep absolute values in logarithms?
Worked answer and reason
The derivative ln|u| is u′/u where u≠0.
Intervals may lie on either side of a root.
Catch the tempting mistake
- Solving A and B at roots allows integrating through those roots.
- The identity is an algebraic tool. The original integrand is still undefined there; an interval crossing a pole needs improper-integral analysis.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
This calculator requires JavaScript. Use the physics formula sheets and show your working manually.
+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 6 problem 24 · No calculator · analytic
- Unit 6 problem 25 · No calculator · analytic
- Unit 6 problem 30 · No calculator · analytic
For the teacher
- Have students multiply the decomposition back together before integrating. Ask them to mark excluded x-values on a number line.
Glossary and deeper reading · 3 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Rational function
A rational function is a quotient of two polynomials. Its domain excludes inputs that make the original denominator zero. For example, (x²−1)/(x−1) agrees with x+1 away from x=1, but remains undefined there. Canceling a common factor does not automatically restore an excluded input.
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Logarithm
A logarithm gives the exponent required to obtain a number from a specified base. For real logarithms, the argument is positive and the base is positive and different from 1. For example, log₂8=3 because 2³=8, while ln uses the base e. Logarithm rules convert products into sums and powers into constant multiples, subject to the domain restrictions.
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Interval
An interval is a set of real numbers containing every number between any two of its members. Parentheses exclude an endpoint, while square brackets include it. For example, (1,4] includes 4 and every number between 1 and 4, but excludes 1. An infinite endpoint always uses a parenthesis because infinity is not a real endpoint.