Antiderivative
In short: Antiderivative of a function f is any function F whose derivative equals f, so finding one means undoing differentiation, and each answer is only fixed up to an added constant.
AP Calculus ABCalculusAP Calculus AB Unit 6
What is Antiderivative?
An antiderivative of \(f\) is a function \(F\) that differentiates back to \(f\): \(F^{\prime}(x) = f(x)\). If a derivative turns position into velocity, an antiderivative turns velocity back into position.
There is never just one. Because constants vanish when you differentiate, \(x^2\), \(x^2 + 7\) and \(x^2 – 100\) are all antiderivatives of \(2x\). Their graphs are the same curve slid up or down, and the whole family is written \(F(x) + C\). An extra fact, such as a known starting value, pins down the constant.
How to spot it
- You are given a derivative or rate and asked to recover the original function.
- The notation is an integral sign without limits, or the wording is “find F(x) such that F'(x) = …”.
- A velocity or acceleration function is paired with a starting position or starting speed.
- An initial condition like F(1) = 10 is supplied to determine the constant of integration.
The method
- Rewrite the function as a sum of powers of x with constant multipliers.
- Reverse the power rule on each term: raise the exponent by 1 and divide by the new exponent.
- Add \(+C\) to the whole result.
- If an initial condition is given, substitute it to solve for \(C\), then check by differentiating.
Antiderivative: solved example
Problem
Find the function \(F\) with \(F^{\prime}(x) = 6x^2 – 4x + 5\) and \(F(1) = 10\).
Solution
Reverse the power rule term by term: \(F(x) = 2x^3 – 2x^2 + 5x + C\).
Use the condition: \(F(1) = 2 – 2 + 5 + C = 5 + C = 10\), so \(C = 5\).
\(F(x) = 2x^3 – 2x^2 + 5x + 5\). Check: its derivative is \(6x^2 – 4x + 5\).
Try it
Question 1
A bike’s velocity is \(v(t) = 12t – 3\) feet per second and its position at \(t = 0\) is 10 feet. Find its position at \(t = 2\).
Show the answer
\(s(t) = 6t^2 – 3t + C\), and \(s(0) = 10\) gives \(C = 10\). Then \(s(2) = 24 – 6 + 10 = 28\) feet.
Question 2
Find the antiderivative of \(\cos x\) whose value at \(x = 0\) is 3.
Show the answer
\(F(x) = \sin x + C\). Since \(\sin 0 = 0\), \(C = 3\), so \(F(x) = \sin x + 3\).
Antiderivative vs. Derivative
The derivative and the antiderivative are opposite operations. Going forward, the answer is one function. Going backward, the answer is a whole family \(F(x) + C\), which is why forgetting \(+C\) costs points.
Keep going
Lessons
Worksheets and quizzes
- Fundamental Theorem of Calculus and Antiderivatives – Worksheet
- Fundamental Theorem of Calculus and Antiderivatives – Quiz
Related words
Whole dictionary: A–Z · AP Calculus AB · Calculus