Module 5 · Lesson 5.1

Antiderivatives — Undoing the Derivative

Given a rate of change, learn how to reconstruct the original function it came from — and why that answer is never quite alone.

Intuition

You already know how to run the derivative machine forward: feed it a function, and it hands back the rate of change. Antiderivatives ask the opposite question. You're handed the rate of change, and you have to figure out what function it came from.

Core concept

A function $F$ is an antiderivative of $f$ if $F'(x) = f(x)$. But here's the catch: if $F(x)$ works, so does $F(x) + 5$, or $F(x) - 12$, or $F(x)$ plus any constant at all — adding a constant never changes the slope. So running the machine backward doesn't give you one answer. It gives you an entire family of them, all differing by a constant.

velocity → position v(t) = 6t s(t) = 3t² + C
Given ratev(t) = 6t
Reverse the power ruleraise power: t → t²
Match the coefficient3t² (since d/dt[3t²] = 6t)
Attach the constants(t) = 3t² + C
Use s(0) = 5 to pin CC = 5
Answers(t) = 3t² + 5

Interactive Lab

Drag the slider to change the constant $C$. Watch the curve slide up and down — and notice that the tangent line's steepness at the marked point never changes. Every curve in this family has the exact same derivative.

x y
F(x) = 0.5x² + 0 F'(x) = x (constant, no matter what C is)

Key Idea

$$ \int x^{n}\,dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1) $$

This is the power rule for antiderivatives — the derivative power rule, run in reverse. Raise the exponent by one, divide by the new exponent, and always tack on $C$, since a constant term vanishes under differentiation and has to be recovered by hand.

$\int$The integral sign — "find the antiderivative of what follows"
$x^{n}$The rate-of-change function you're starting from
$dx$Marks $x$ as the variable being integrated over
$C$The constant of integration — stands in for the entire family of valid answers

Think Further

  1. Two students each find an antiderivative of $f(x) = 4x$. One gets $2x^2 + 1$, the other gets $2x^2 - 7$. Can they both be correct? How would you check?
  2. You're told an object's velocity but not its starting position. Can you find its exact position at every later time? What extra piece of information would you need?
  3. The power rule for antiderivatives explicitly excludes $n = -1$. Looking at the formula, predict what goes wrong if you tried to plug in $n = -1$ anyway.
Show suggested answers
  1. Yes — both are valid antiderivatives of $4x$, since they differ only by a constant. Differentiating either one gives back $4x$, which is the real test: check by differentiating, not by comparing the two answers directly.
  2. Not exactly — without an initial position (like $s(0)$), you only know the shape of the position function up to a vertical shift. You need one known data point to pin down the specific constant $C$.
  3. The formula divides by $n+1$, which becomes division by zero when $n = -1$. The rule breaks down there, and $x^{-1}$ needs a different antiderivative (it turns out to involve the natural logarithm, which is covered later in the course).