What it is
F is an antiderivative of f on an interval if F′(x) = f(x) for every x in the interval. You look for a function that gives f when it is differentiated, and you check the answer by differentiating it.
- x² is an antiderivative of 2x, because the derivative of x² is 2x.
- So are x² + 5 and x² − 3. A constant has derivative 0 and leaves no trace after differentiating. On an interval, two antiderivatives of the same f differ only by a constant, so all of them can be written x² + C. The number C is called the constant of integration.
- ∫ 2x dx = x² + C is the notation for the whole family. It is called the indefinite integral of 2x.
A few pairs worth knowing. The right-hand column is the check.
| f(x) | An antiderivative F(x) | F′(x) |
|---|---|---|
| 3x² | x³ | 3x² |
| x | x²/2 | x |
| 5 | 5x | 5 |
| cos x | sin x | cos x |
| eˣ | eˣ | eˣ |
| 1/x, x > 0 | ln x | 1/x |
Not antiderivatives:
- x² for f(x) = x. The derivative of x² is 2x, not x. An antiderivative of x is x²/2.
- 2x for f(x) = x². The derivative of 2x is 2. An antiderivative of x² is x³/3.
A function that is continuous on an interval has an antiderivative there, but it cannot always be written as a formula. e^(−x²) has an antiderivative, but no formula for it built from powers, roots, exponentials, logarithms and trigonometric functions.
Why it matters
A definite integral is computed with an antiderivative. If f is continuous on [a, b] and F is an antiderivative of f, then ∫ₐᵇ f(x) dx = F(b) − F(a); see definite integral. With F(x) = x², ∫₁³ 2x dx = 3² − 1² = 8. The choice of antiderivative does not change the answer, because C appears in both F(b) and F(a) and cancels. Why this works is the fundamental theorem of calculus.
- Differential equations. The solutions of y′ = f(x) are the antiderivatives of f, y = F(x) + C; see the differential equation y′ = f(x). A condition picks one of them: y′ = 2x and y(0) = 5 give y = x² + C with C = 5, so y = x² + 5.
- Motion. Position is an antiderivative of velocity, just as velocity is the derivative of position. With v(t) = 3t² m/s and a position of 5 m at t = 0, s(t) = t³ + 5, and after 2 s the position is 13 m.
- The rules for the common functions are the differentiation rules read backwards: powers, sin x, cos x, 1/x, eˣ and constants.
The integral calculator finds antiderivatives, and the derivative calculator checks them.
Common misconceptions
- “There is only one antiderivative.” There are infinitely many, and on an interval they differ by a constant. That is what + C is for in the general answer. In a definite integral the constant cancels.
- “An antiderivative and an integral are the same thing.” F is a function. The indefinite integral ∫ f(x) dx is the whole family F(x) + C. The definite integral ∫ₐᵇ f(x) dx is a number, 8 in the example above.
- “For a product, multiply the antiderivatives.” An antiderivative of x · x = x² is x³/3. The product (x²/2) · (x²/2) = x⁴/4 has derivative x³, not x².
- “ln x is the antiderivative of 1/x, full stop.” That holds for x > 0. 1/x is also defined for x < 0, where ln x is not, and there ln(−x) is an antiderivative. Both cases are written ln|x|; see antiderivative of 1/x.
Common questions
How do I check that F is an antiderivative of f?
Differentiate F. If you get f, F is an antiderivative of f. F(x) = x³/3 + 2x has derivative x² + 2, so it is an
antiderivative of x² + 2.
Why is + C sometimes written and sometimes not?
The answer to an indefinite integral is a family of functions, and C stands for any member of it. The answer to
a definite integral is a number, and there C cancels. A condition gives C a value: F(x) = x² + C and F(0) = 5
give C = 5.
Does every function have an antiderivative?
Every function that is continuous on an interval has one. A function with a jump does not: f(x) = 0 for x < 1
and f(x) = 1 for x ≥ 1 has no antiderivative on any interval with points on both sides of x = 1. And where an
antiderivative exists, a formula for it may not, as for e^(−x²). A definite integral is then computed
numerically, for example with the trapezoidal rule.