Antiderivatives and the Indefinite Integral
An antiderivative of \(f\) is a function \(F\) whose derivative is \(f\). The indefinite integral \(\int f(x)\,dx\) collects every antiderivative at once — a whole family of curves differing only by the constant \(+C\).
By the end you'll be able to antidifferentiate any polynomial with the reverse-power, constant-multiple, and sum rules, explain why the answer always carries \(+C\), and check your result by differentiating it.
Predict: as you slide C, does the slope of the tangent at x = 1 change? Drag the C slider and check.
Every curve below is \(F(x) + C = 2x^3 + 2x^2 - 5x + C\) for a different constant — the whole family written \(\int (6x^2 + 4x - 5)\,dx\). Slide C to move the bright curve through the stack, and slide x to move the tangent points. The tangent slope at your chosen x is the same on every curve: it's \(f(x) = 6x^2 + 4x - 5\), no matter what C is. Hover or tab to any dot for exact values.
Tangent slope at x = 1.0 on every curve: f(1.0) = 5.00 · bright curve: C = 0.0, F(1.0) + C = -1.00
Antidifferentiation asks the derivative question backward: "what function did I differentiate to get this?" Because infinitely many functions share the same slope everywhere — they differ only by a vertical shift — the answer is always a family, never a single function.
\(F\) is an antiderivative of \(f\) when \(F'(x) = f(x)\). Since the derivative of a constant is zero, \(F(x) + C\) is also an antiderivative for every constant \(C\) — and the indefinite integral packs that whole family into one symbol: \[\int f(x)\,dx = F(x) + C,\] where \(C\) is the constant of integration. Unlike a definite integral, an indefinite integral is not a number — it's a set of functions: the stack of parallel curves you just dragged through, all sharing the same tangent slope \(f(x)\) at every x.
For polynomials, three rules do all the work. The reverse power rule undoes the power rule for derivatives — raise the exponent by one, then divide by the new exponent: \(\int x^n\,dx = \tfrac{x^{n+1}}{n+1} + C\) (for \(n \neq -1\)). The constant-multiple rule lets coefficients ride along, and the sum rule lets you integrate term by term, adding a single \(+C\) at the end. Best of all, every answer is self-checking: differentiate it, and you must get the integrand back.
Find \(\displaystyle\int (6x^2 + 4x - 5)\,dx\) — the same integrand as the interactive.
- Reverse power rule on each power: raise the exponent, divide by the new one. \(6x^2 \to 6 \cdot \tfrac{x^3}{3} = 2x^3\); \(4x \to 4 \cdot \tfrac{x^2}{2} = 2x^2\).
- Constant term: \(-5 = -5x^0\) antidifferentiates to \(-5x\).
- Sum rule: add the pieces with one constant: \(\int (6x^2 + 4x - 5)\,dx = 2x^3 + 2x^2 - 5x + C\).
- Check by differentiating: \(\tfrac{d}{dx}(2x^3 + 2x^2 - 5x + C) = 6x^2 + 4x - 5\). ✓
Find \(\displaystyle\int (9x^2 - 2x + 4)\,dx\). Reverse power rule: \(9x^2 \to 9 \cdot \tfrac{x^3}{3} = 3x^3\), and \(-2x \to -2 \cdot \tfrac{x^2}{2} = -x^2\). Now finish it: the constant term \(4\) becomes ____, so \(\int (9x^2 - 2x + 4)\,dx =\) ____
Reveal the answer
\(4 = 4x^0\) antidifferentiates to \(4x\), so \(\int (9x^2 - 2x + 4)\,dx =\) \(3x^3 - x^2 + 4x + C\). Check: \(\tfrac{d}{dx}(3x^3 - x^2 + 4x + C) = 9x^2 - 2x + 4\). ✓ Don't drop the \(+C\) — the answer is the whole family.
More info — why +C is the entire difference between antiderivatives
Suppose \(F\) and \(G\) are both antiderivatives of the same \(f\). Then \((F - G)' = f - f = 0\) — their difference has zero slope everywhere, so it must be a constant. That's exactly what the interactive shows from the other direction: shifting a curve by C never changes any tangent slope, and no other change is possible without changing the derivative. So \(+C\) isn't optional bookkeeping; it names every solution there is. The OpenStax chapter in Dive deeper below proves this and extends it to picking out ONE curve when you know a point it must pass through.
More info — where the reverse power rule breaks
Two warnings from the trenches. First, \(\int x^{-1}\,dx\) is NOT \(\tfrac{x^0}{0}\) — that's division by zero. The \(n = -1\) case is a logarithm, which you'll meet with the standard integral forms later in this section. Second, there is no product or quotient rule for integrals — you can't integrate a product factor by factor; rewrite the integrand first, or wait for substitution.
Check your understanding
Compute \(\int (12x^3 - 6x + 7)\,dx\).
Why must an indefinite integral include \(+C\)?
Use the power rule for derivatives to check: which function is an antiderivative of \(x^3\)?
The reverse power rule \(\int x^n\,dx = \tfrac{x^{n+1}}{n+1} + C\) fails for exactly one of these integrands. Which one?
Recap
- \(F\) is an antiderivative of \(f\) when \(F'(x) = f(x)\); the indefinite integral \(\int f(x)\,dx = F(x) + C\) names the whole family, not one function.
- Any two antiderivatives of the same \(f\) differ only by a constant — vertically shifted curves with identical (parallel) tangents at every x.
- Reverse power rule: \(\int x^n\,dx = \tfrac{x^{n+1}}{n+1} + C\) for \(n \neq -1\) — raise the exponent, divide by the new exponent.
- Constant-multiple and sum rules: integrate polynomials term by term, coefficients ride along, one \(+C\) at the end.
- Always check by differentiating: your answer's derivative must be the integrand.
Dive deeper
- OpenStax Calculus Volume 1 — 4.10 Antiderivatives Define antiderivatives and the indefinite integral with the power rule for integrals
Sources
- Antiderivatives and the Indefinite Integral