When should we divide before looking for an antiderivative?
You will be able to: Rewrite rational functions by polynomial division, preserving the domain.
When should we divide before looking for an antiderivative?
Eleven objects divided into groups of four make two full groups with three left over. Polynomial division similarly separates a whole-polynomial part from a smaller remainder.
A useful starting point: When changing variables, what happens to the endpoints? →
Words and symbols before equations
- Rational function
- A quotient of polynomials.
- Degree
- Highest power with a nonzero coefficient.
- Quotient and remainder
- The polynomial part and leftover part after division.
- Proper fraction
- Numerator degree smaller than denominator degree.
What this picture assumes
Original model; readouts are rounded. f=(x²+1)/(x+1)=x−1+2/(x+1). F=x²/2−x+2ln(x+1) on x>−1. Model displays [0,3]; x=−1 is excluded from the original expression.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- Original=1; quotient-plus-remainder=1; F=0.886294. Division preserves equality at every allowed input.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
When a polynomial numerator has degree at least that of its denominator, division may reveal easy integral forms. For (x²+3x+1)/(x+1), the quotient is x+2 and remainder −1.
Verify by multiplying: (x+1)(x+2)−1=x²+3x+1. Thus the integrand is x+2−1/(x+1), for x≠−1.
Integrating term by term gives x²/2+2x−ln(abs(x+1))+C. The logarithm comes from the remaining reciprocal linear factor.
Algebraic rewriting does not erase excluded inputs. Never apply endpoint FTC across a denominator zero as if the function were continuous; improper integrals belong to BC topic 6.13 and are outside this AB unit.
A worked example, step by step
Find ∫(x²+1)/(x+1)dx on x>−1.
- Divide: x²+1=(x+1)(x−1)+2.
- Rewrite the integrand as x−1+2/(x+1).
- Integrate to x²/2−x+2ln(x+1)+C on x>−1.
- Differentiate and combine over x+1 to recover (x²+1)/(x+1).
Do not integrate numerator and denominator separately and divide the answers. Rewrite the integrand itself with a valid identity.
Does division allow x=−1 in (x²+1)/(x+1)?
Compare with an explanation
No. The original denominator vanishes there; the rewritten form must preserve that exclusion.
Predict. Change one thing. Explain.
Move x through the displayed positive interval. Compare the original rational value with the quotient-plus-remainder value and the derivative of the proposed antiderivative.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Original=1; quotient-plus-remainder=1; F=0.886294. Division preserves equality at every allowed input.
Original model; readouts are rounded. f=(x²+1)/(x+1)=x−1+2/(x+1). F=x²/2−x+2ln(x+1) on x>−1. Model displays [0,3]; x=−1 is excluded from the original expression.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the rate units, signed contributions, theorem conditions, domain or antiderivative check. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionIntegrate (x²+2x+2)/(x+1) on x>−1 and check the algebra and derivative.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: x²+2x+2=(x+1)(x+1)+1.
- 1 point: The integrand becomes x+1+1/(x+1).
- 1 point: An antiderivative is x²/2+x+ln(x+1)+C.
- 1 point: Its derivative is x+1+1/(x+1), matching the original quotient on the allowed domain.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1Why perform division?
To expose simpler terms with recognizable antiderivatives.
RECALL 2What happens to the remainder?
It stays over the original denominator.
RECALL 3May a domain exclusion be forgotten after rewriting?
No.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
When should we divide before looking for an antiderivative?
- Polynomial numerator=denominator×quotient+remainder.
- Divide when numerator degree≥denominator degree, then inspect the remainder.
- Preserve excluded denominator zeros.
Remember: Do not integrate numerator and denominator separately and divide the answers. Rewrite the integrand itself with a valid identity.
Conditions: Original model; readouts are rounded. f=(x²+1)/(x+1)=x−1+2/(x+1). F=x²/2−x+2ln(x+1) on x>−1. Model displays [0,3]; x=−1 is excluded from the original expression.
Refresh Kid · AP Calculus BC Unit 6 · Objectives FUN-6.D · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 6.10, FUN-6.D. CED effective Fall 2020 and Fall 2026 clarifications checked September 17, 2026. BC scope includes all fourteen topics, including integration by parts (6.11), nonrepeating linear partial fractions (6.12) and improper integrals (6.13). Topic 6.14 integrates the BC toolkit and Skill 1.C. Focused lesson titles, examples and questions are original Refresh Kid material.
Distinguish signed accumulation, unsigned area and initial amount. Error direction needs interval-wide shape information. FTC hypotheses, variable-bound chain factors, integration constants, transformed bounds and domain restrictions are explicit. Bounded jumps are treated separately from differentiability of accumulation. The BC extension teaches parts from the product rule, decomposition with distinct linear factors and independent improper limits. Finite truncations do not certify convergence. Shared foundation lessons are maintained with AB; BC extensions and method selection are authored for this course.
The Organic Chemistry Tutor Fundamental Theorem of Calculus Part 1 video title, creator and description were checked; the full video was not reviewed. Khan Academy’s unit destination was checked, but its JavaScript lesson contents were not fully readable by the research tool. OpenStax Sections 5.2, 5.3 and 5.5 were consulted for conceptual cross-checking. No provider scripts, questions, diagrams or artwork were copied. Refresh Kid is not affiliated with or endorsed by these providers.
GitHub’s 3D website collection informed optional spatial inspection and camera controls. The original tank geometry uses existing self-hosted Three.js with its MIT license. Its 2×2 dm base and water depth use the same volume as the rate calculation: 1 dm³=1 L. The initial water is blue and newly accumulated inflow is teal. The rate graph is an abstract amount calculation; it is not the physical shape of water. Camera rotation changes only the view. Full 2D graphs, readouts and explanations remain available without WebGL.
Independent teacher review and observation of students remain pending. Technical checks do not certify mathematical accuracy, accessibility or learning effectiveness. This is a review edition.
Optional official resource: Released AP Calculus BC questions and scoring guides. The archive spans multiple units; no entire exam question is assigned here.
Learn → Explore → Practice → Review is informed by the IES learning guide; this exact implementation has not been evaluated with learners.
Want to work through this with a tutor?
Bring your question about When should we divide before looking for an antiderivative? Your explanation and answers remain free to access.