Integral Calculator
Find antiderivatives and definite integrals, with the area under the curve shaded on a graph.
Press Enter to solve · ^ for powers · sqrt() for roots · separate a system with a comma
The integral calculator finds antiderivatives (indefinite integrals) and evaluates definite integrals. For an area between limits, type x^2 from 0 to 3; the tool applies the Fundamental Theorem of Calculus and shades the area on the graph.
How to use the integral calculator
- Enter the function, e.g.
x^2 + 3xorsin(x). - For a definite integral add the limits:
from a to b. - Press Enter to see the antiderivative, the evaluation F(b) − F(a) and the graph.
Basic integration rules
| Function | Integral |
|---|---|
| xⁿ (n ≠ −1) | xⁿ⁺¹ / (n + 1) + C |
| 1/x | ln|x| + C |
| eˣ | eˣ + C |
| sin x | −cos x + C |
| cos x | sin x + C |
Example: ∫₀³ x² dx
- Antiderivative: F(x) = x³/3.
- F(3) − F(0) = 27/3 − 0.
- Result: 9 square units.
Questions about integrals
What is the difference between definite and indefinite integrals?
An indefinite integral is a family of functions, F(x) + C. A definite integral is a number: the signed area under the curve between two limits.
What if there is no simple antiderivative?
For a definite integral the calculator falls back to Simpson’s rule and gives a numerical answer accurate to six decimal places.
Can a definite integral be negative?
Yes. Area below the x-axis counts as negative, so a definite integral can be negative or zero.