How do we reverse differentiation without losing the constant?
You will be able to: Find power-rule antiderivatives and explain the family represented by +C.
How do we reverse differentiation without losing the constant?
Two cyclists can have identical speeds but different starting positions. Their position graphs can stay a fixed distance apart. Knowing a derivative similarly leaves an unknown vertical shift.
A useful starting point: Why do antiderivative endpoint values give the integral? →
Words and symbols before equations
- Indefinite integral
- The family of antiderivatives, written without fixed bounds.
- C
- An arbitrary constant of integration on an interval.
- Power n
- The exponent in xⁿ.
- Domain interval
- A connected interval on which the formula is defined.
What this picture assumes
Original model; readouts are rounded. F=x²+C and F′=2x on all real x. Display window −2≤x≤2. A known value F(1)=5 selects C=4; the family otherwise has arbitrary C.
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.
- C=0; F(1)=1; F′(x)=2x for every C. The condition F(1)=5 requires C=4.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Since d/dx[xⁿ⁺¹/(n+1)]=xⁿ for n≠−1, ∫xⁿdx=xⁿ⁺¹/(n+1)+C wherever the real power is defined on the interval. Increase the exponent by one, then divide by that new exponent.
Integrate sums term by term and pull out constant multipliers. For ∫(6x²−4x+3)dx, the result is 2x³−2x²+3x+C.
Every constant shift has the same derivative, so one antiderivative is not the complete family. On disconnected domains, different constants may apply on different intervals.
The exponent −1 is exceptional: dividing by n+1 would divide by zero. Its integral is ln(abs(x))+C on an interval excluding zero, as explored in the next lesson.
| Feature | Definite integral | Indefinite integral |
|---|---|---|
| Bounds | Specified | No fixed endpoint bounds |
| Result | A signed number | An antiderivative family |
| Constant | Cancels on evaluation | Include +C on an interval |
A worked example, step by step
Find ∫(4x³+2√x)dx for x>0 and verify it.
- Rewrite √x as x^(1/2).
- The first term integrates to x⁴.
- The second integrates to 2x^(3/2)/(3/2)=(4/3)x^(3/2).
- The family is x⁴+(4/3)x^(3/2)+C; differentiating gives 4x³+2√x.
Integration increases the power and divides by the new power. It is not the differentiation rule in reverse order without changing the exponent.
Why do x² and x²+7 have the same derivative?
Compare with an explanation
The derivative of a constant is zero, so their slopes agree everywhere.
Predict. Change one thing. Explain.
Move the constant C in F=x²+C while the derivative f=2x stays fixed. Explain what changes about the graph and what does not change about its slope.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
C=0; F(1)=1; F′(x)=2x for every C. The condition F(1)=5 requires C=4.
Original model; readouts are rounded. F=x²+C and F′=2x on all real x. Display window −2≤x≤2. A known value F(1)=5 selects C=4; the family otherwise has arbitrary C.
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 questionFind the antiderivative family of 9x²−2/x² for x>0, and verify each term.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Rewrite −2/x² as −2x^(−2).
- 1 point: The first term integrates to 3x³.
- 1 point: The second integrates to 2x^(−1)=2/x.
- 1 point: F=3x³+2/x+C has derivative 9x²−2/x² on x>0.
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 1What does +C represent?
All constant vertical shifts of an antiderivative on an interval.
RECALL 2What happens to a constant integrand?
Its antiderivative is the constant times x.
RECALL 3Why exclude n=−1?
The power-rule denominator would be zero.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do we reverse differentiation without losing the constant?
- ∫xⁿdx=xⁿ⁺¹/(n+1)+C, n≠−1, on a valid domain interval.
- ∫k dx=kx+C.
- Check by differentiating.
Remember: Integration increases the power and divides by the new power. It is not the differentiation rule in reverse order without changing the exponent.
Conditions: Original model; readouts are rounded. F=x²+C and F′=2x on all real x. Display window −2≤x≤2. A known value F(1)=5 selects C=4; the family otherwise has arbitrary C.
Refresh Kid · AP Calculus BC Unit 6 · Objectives FUN-6.C · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 6.8, FUN-6.C. 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.
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