Quick start
- Type f(x), for example 6x^2 - 4x + 1 or sin(x).
- Enter the lower limit a and the upper limit b.
- Read F(x) + C and F(b) − F(a) for a polynomial, or the numerical value for anything else.
Antiderivatives with the power rule
Integration reverses differentiation. For each term, raise the power by one and divide by the new power: ∫xⁿ dx = xⁿ⁺¹ ÷ (n + 1) + C. So ∫(6x² − 4x + 1) dx = 2x³ − 2x² + x + C.
Example: from 1 to 3
With F(x) = 2x³ − 2x² + x, F(3) = 54 − 18 + 3 = 39 and F(1) = 2 − 2 + 1 = 1. The definite integral is 39 − 1 = 38.
Signed area
A definite integral counts area above the x-axis as positive and area below it as negative. The integral of sin(x) from 0 to 2π is 0, because the two humps cancel.
Non-polynomial functions
For functions like eˣ, 1/x or sin(x), the tool uses Simpson's rule with 2,000 slices. That is accurate to many decimal places for smooth curves. It warns you if the function breaks down inside the interval, such as 1/x across 0.
Common questions
Why is there a + C?
Any constant disappears when you differentiate, so an antiderivative is only known up to a constant.
What happens if I swap the limits?
The answer changes sign.
Does it use radians?
Yes, as calculus requires.