An antiderivative is a function whose derivative gives back the function you started with. It is the reverse process of differentiation, so it helps connect rates of change to original quantities. Antiderivatives matter in physics, engineering, and math because many problems give a rate and ask for the accumulated result.
For example, if velocity is known as a function of time, an antiderivative gives position up to an unknown starting location.
A single function usually has infinitely many antiderivatives because adding a constant does not change the derivative. This creates a family of vertically shifted curves, all with the same slope pattern at matching x-values. The constant of integration, written C, represents that unknown vertical shift or starting value.
Basic antiderivative rules let you build more complicated antiderivatives from powers, constants, sums, and common functions.
Key Facts
- If F'(x) = f(x), then F(x) is an antiderivative of f(x).
- The general antiderivative is written ∫ f(x) dx = F(x) + C.
- Power rule for antiderivatives: ∫ x^n dx = x^(n+1)/(n+1) + C, for n ≠ -1.
- Constant multiple rule: ∫ k f(x) dx = k ∫ f(x) dx.
- Sum rule: ∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx.
- Special case: ∫ 1/x dx = ln|x| + C, for x ≠ 0.
Vocabulary
- Antiderivative
- An antiderivative of f(x) is a function F(x) whose derivative is f(x).
- Indefinite integral
- An indefinite integral is notation for the entire family of antiderivatives of a function.
- Constant of integration
- The constant of integration C represents any vertical shift that does not change the derivative.
- Family of functions
- A family of functions is a set of related functions, such as F(x) + C, that differ by a constant.
- Initial condition
- An initial condition is a known value of a function used to find the specific constant C.
Common Mistakes to Avoid
- Forgetting + C is wrong because an indefinite integral represents all antiderivatives, not just one curve.
- Using the power rule on ∫ 1/x dx is wrong because the formula ∫ x^n dx = x^(n+1)/(n+1) + C does not work when n = -1.
- Adding exponents instead of increasing by one and dividing is wrong because ∫ x^n dx requires both steps, giving x^(n+1)/(n+1) + C.
- Treating C as a slope change is wrong because adding a constant shifts the graph vertically but leaves every derivative value unchanged.
Practice Questions
- 1 Find the general antiderivative of f(x) = 6x^2 - 4x + 9.
- 2 A particle has velocity v(t) = 3t^2 + 2t meters per second. Find the position function s(t) if s(0) = 5 meters.
- 3 Explain why the graphs of y = x^3 + 2, y = x^3 - 4, and y = x^3 + 10 all have the same derivative.