How to use it
- Type the function to integrate, for example x*e^x.
- Choose Indefinite or Definite integral.
- For a definite integral, enter the lower and upper limits.
- Click Integrate and read the result.
Indefinite integrals (antiderivatives)
An indefinite integral F(x) is a function whose derivative is f(x); it's written with + C because any constant works. The calculator finds F symbolically and then checks it by differentiating: if F′(x) doesn't match f(x) numerically, the result isn't shown.
Example: ∫ x·eˣ dx = (x − 1)·eˣ + C. Check: the derivative of (x − 1)eˣ is eˣ + (x − 1)eˣ = x·eˣ.
When the result contains ln, read it with an absolute value where the argument can be negative: ∫ 1/x dx = ln|x| + C.
Definite integrals
A definite integral ∫ₐᵇ f(x) dx is the signed area under the curve between a and b. When an antiderivative exists, the calculator uses the Fundamental Theorem of Calculus, F(b) − F(a), to give an exact value, and confirms it numerically. If there is no elementary antiderivative, it still computes the value with adaptive Simpson's rule, to about 10 decimal places.
Example: ∫₀¹ x·eˣ dx = [(x − 1)eˣ]₀¹ = 0 − (−1) = 1.
Limits can be numbers or expressions such as pi/2. If the function is undefined or infinite inside the interval (for example 1/x across 0), the calculator says the integral can't be computed this way.
Common integrals
| f(x) | ∫ f(x) dx |
|---|---|
| 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 |
| 1/(1 + x²) | arctan x + C |
| 1/√(1 − x²) | arcsin x + C |
Some functions, like e^(−x²) or sin(x)/x, have no antiderivative in elementary functions. For those the calculator explains that and still gives definite integrals numerically, which is how the normal distribution's probabilities are computed, for example.
How to enter functions
Type functions the way you'd write them: x^3, sqrt(x) or √x, 2x or 3*x, sin(x)^2 or sin^2(x), e^x or exp(x), ln(x) for the natural log, log10(x), abs(x), and the constants pi and e. Implicit multiplication like 2x, x(x+1) and sinx is understood.
Tips
- Results can look different from your textbook but be equal, or differ only by a constant; both are correct antiderivatives.
- For a definite integral of a function that changes sign, the result is the net (signed) area. For total area, split the interval where f(x) = 0 and add absolute values.
- The calculation runs in your browser; nothing is sent to a server.
Frequently asked questions
What is the integral of x*e^x?
(x − 1)·eˣ + C, found by integration by parts.
Can it calculate definite integrals?
Yes. Choose Definite integral and enter the limits. You get an exact value when possible and a numerical value otherwise.
Why is there no antiderivative for some functions?
Some functions, such as e^(−x²), have no antiderivative expressible with elementary functions. Their definite integrals can still be computed numerically.
Does the integral calculator show steps?
It shows the antiderivative and verifies it by differentiation, and for definite integrals the F(b) − F(a) evaluation; it doesn't list integration techniques step by step.
How accurate are numerical results?
About 10 decimal places with adaptive Simpson's rule, for functions that are well behaved on the interval.
Last updated: November 12, 2026