The constant of integration is the arbitrary constant (+C) added to every indefinite integral, because infinitely many functions share the same derivative. On the AP Calculus exam, it turns one antiderivative into a whole family of general solutions, and an initial condition picks out the specific one.
When you antidifferentiate, you're running differentiation in reverse. The problem is that differentiation destroys information. The derivative of x² + 5 is 2x, but so is the derivative of x² − 100 and x² + π. Any constant vanishes when you differentiate. So when you go backward and write ∫2x dx, you can't recover which constant was there. You write x² + C to represent every possible answer at once.
That +C isn't decoration. It means an indefinite integral is a family of functions, all vertical shifts of each other, every one with the same derivative. Graphically, picture infinitely many parallel curves stacked vertically. In Unit 7, this is exactly why separable differential equations give you a general solution first. The +C is the placeholder, and an initial condition like y(0) = 91 is what nails down a single curve, the particular solution.
The constant of integration lives at the intersection of Unit 6 (Integration and Accumulation of Change) and Unit 7 (Differential Equations). It shows up the moment you compute any indefinite integral in Topic 6.14, and it becomes the whole point in Topic 7.6, where learning objective AP Calc 7.6.A asks you to determine general solutions to differential equations using antidifferentiation and separation of variables. The CED's essential knowledge for 7.6.A is blunt about it. Antidifferentiation produces general solutions, and general solutions only exist because of +C. Forgetting it on an FRQ doesn't just cost the point where you dropped it. It usually wrecks the follow-up steps, because you can't apply the initial condition correctly without it.
Keep studying AP Calculus Unit 6
Visual cheatsheet
view galleryGeneral Solution (Unit 7)
The general solution IS the constant of integration in action. When you separate variables and integrate both sides, the +C is what makes your answer a family of curves instead of one curve. Solve for C using an initial condition and the general solution collapses into a particular solution.
Definite Integral (Unit 6)
Definite integrals never need a +C. The Fundamental Theorem computes F(b) − F(a), and any constant you'd add cancels in the subtraction. That cancellation is the quickest way to see why +C belongs only to indefinite integrals.
Limits of Integration (Unit 6)
Improper integrals (Topic 6.13, learning objective AP Calc 6.13.A) are evaluated as limits of definite integrals. Since they're built from definite integrals, the constant of integration drops out there too. The antiderivative is a stepping stone, not the final answer.
Integration by Parts (Unit 6, BC)
Every antidifferentiation technique, from u-substitution to integration by parts, ends the same way. Once the integral sign is gone, +C goes on. A common slip in multi-step techniques is adding C in the middle of the work and losing track of it. Add it once, at the end.
Differential equation FRQs are where +C earns or loses you real points, and one shows up almost every year. The 2017 (cooling potato), 2018 (dy/dx = x(y − 2)²), 2019 (draining barrel), and 2023 (warming milk) FRQs all required separating variables, antidifferentiating, writing +C, and then using an initial condition to solve for it. The scoring guidelines specifically award a point for the constant of integration and the initial condition together, and if you omit +C, you're typically not eligible for the remaining solution points. Multiple-choice questions test the concept more quietly. A stem might ask for the general solution of dy/dx = 4eˣ (answer: y = 4eˣ + C) and put the C-less version among the wrong choices, or ask what an initial condition determines (it determines the value of C, turning the general solution into a particular one). One more habit worth building, when you exponentiate both sides of ln|y| = x + C, the constant transforms, since e^(x+C) becomes Aeˣ where A = e^C.
An indefinite integral asks 'what family of functions has this derivative?' so it needs +C to represent every member. A definite integral asks 'what number does this accumulation equal?' and the Fundamental Theorem subtracts F(b) − F(a), which cancels any constant. Rule of thumb. Limits of integration on the integral means no +C; no limits means +C is mandatory.
Every indefinite integral needs a +C because infinitely many functions, all vertical shifts of each other, share the same derivative.
Definite integrals never include +C, since the constant cancels when you compute F(b) − F(a).
In separation of variables (Topic 7.6), the +C is what makes your answer a general solution, and the initial condition is what you use to solve for C.
Add +C only once, immediately after the last integral sign disappears, even if you integrated both sides of an equation.
When you exponentiate ln|y| = x + C, the constant changes form, so e^(x+C) becomes a new constant A times eˣ.
Forgetting +C on a differential equations FRQ usually costs more than one point, because the initial-condition and particular-solution steps depend on it.
It's the arbitrary constant +C added to every indefinite integral. Since the derivative of any constant is zero, antidifferentiation can't tell which constant the original function had, so +C stands in for all possibilities at once.
No. A definite integral evaluates to a number via F(b) − F(a), and any +C cancels in that subtraction. The constant only matters for indefinite integrals and general solutions to differential equations.
Yes, on differential equation FRQs like 2017 Q4 or 2023 Q3, the constant of integration is tied directly to a scoring point, and omitting it usually makes you ineligible for the particular-solution points that follow. It's one of the most common avoidable point losses on the exam.
The constant of integration is the +C itself; the general solution is the full equation containing it, like y = x² + C. The general solution describes the entire family of curves, and solving for C with an initial condition gives you one particular solution.
An initial condition like y(0) = 91 lets you plug values into the general solution and solve for C. That converts the infinite family of solutions into the single particular solution the FRQ is asking for.