Quick answer
The indefinite integral ∫ f(x) dx = F(x) + C, and the definite integral is F(b) − F(a). For x²eˣ the antiderivative is eˣ(x² − 2x + 2) + C, and from 0 to 1 the definite integral is e − 2 = 0.718282.
Enter a function of x and optionally lower and upper bounds. The calculator returns the antiderivative and the definite integral.
Related tools: Derivative Calculator: Derivative of a function of x, with a numeric check. Limit Calculator: Two-sided limit of a function of x, with a numeric check. Fraction Calculator: Add, subtract, multiply and divide fractions with simplified and mixed-number results.
How to use the Integral Calculator
- Type function of x, for example x^3*sin(x).
- Optionally, enter lower bound (optional).
- Optionally, enter upper bound (optional).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Function of x | Text, such as x^3*sin(x) | — |
| Lower bound (optional) (optional) | A number | — |
| Upper bound (optional) (optional) | A number | — |
Formula and method
Formula
∫ f(x) dx = F(x) + C; ∫ₐᵇ f(x) dx = F(b) − F(a)
Symbolic result; numeric values to six decimals.
The antiderivative is verified by differentiating it numerically and comparing with the original function. Integrals the engine cannot express in closed form are reported as unsupported.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Function of x = x^2*e^x; Lower bound (optional) = 0; Upper bound (optional) = 1.
| Result | Value |
|---|---|
| Antiderivative | e^x*(2-2*x+x^2) + C |
| Definite integral | 0.718282 |
Example 2
Inputs: Function of x = 1/(x^2+1); Lower bound (optional) = 0; Upper bound (optional) = 2.
| Result | Value |
|---|---|
| Antiderivative | atan(x) + C |
| Definite integral | 1.107149 |
Example 3
Inputs: Function of x = sin(x)^2; Lower bound (optional) = 0; Upper bound (optional) = 3.
| Result | Value |
|---|---|
| Antiderivative | (1/2)*x+(-1/4)*sin(2*x) + C |
| Definite integral | 1.569854 |
The first line is the headline result (antiderivative). Below it the integral Calculator lists definite integral, which describe the same case from other angles. Symbolic result; numeric values to six decimals.
Checking by differentiation
Differentiating eˣ(x² − 2x + 2) gives eˣ(x² − 2x + 2) + eˣ(2x − 2) = x²eˣ ✓. This was found by integration by parts twice. Evaluate at 1: e(1 − 2 + 2) = e, at 0: 1 × 2 = 2, so the integral is e − 2.
Scope
The calculator handles elementary functions of one variable x; anything the engine cannot integrate in closed form is rejected rather than approximated. Some functions, such as e^(−x²), have no elementary antiderivative.
How the result changes with upper bound
Holding every other input at the first worked example, this table changes only upper bound. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Upper bound | Antiderivative | Definite integral |
|---|---|---|
| 0.5 | e^x*(2-2*x+x^2) + C | 0.060902 |
| 1 | e^x*(2-2*x+x^2) + C | 0.718282 |
| 2 | e^x*(2-2*x+x^2) + C | 12.778112 |
| 4 | e^x*(2-2*x+x^2) + C | 543.9815 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Function of x: left empty shows "Enter function of x." beside that field and no result.
- Upper bound (optional): left empty shows "Enter both bounds or neither." beside that field and no result.
Choosing between the Integral Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Derivative Calculator](/math/derivative-calculator/) | function of x and order | Derivative |
| [Limit Calculator](/math/limit-calculator/) | function of x and x approaches | Limit |
| [Fraction Calculator](/math/fraction-calculator/) | numerator 1, denominator 1 and operation | Result |
| [Ratio Calculator](/math/ratio-calculator/) | first number and second number | Simplified ratio |
| [Random Number Generator](/math/random-number-generator/) | minimum, maximum and how many | Random numbers |
Background: calculus, vectors and matrices
A derivative measures how fast a function's output changes with its input, and an integral accumulates a quantity such as area. The determinant of a 2×2 matrix [[a, b], [c, d]] is ad − bc, so [[3, 8], [4, 6]] has determinant 3×6 − 8×4 = −14.
The dot product of two vectors is the sum of the products of matching components and is a single number; a positive value means the vectors point in broadly the same direction.
Assumptions and limitations
- Handles elementary functions of one variable x; anything the engine cannot simplify or confirm is reported as unsupported; antiderivatives of 1/x are shown as ln|x| + C in the usual convention
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for function of x and upper bound (optional) to confirm this.
Frequently asked questions
What does the +C mean?
An arbitrary constant, since any constant disappears when differentiating.
What is the area under a curve?
The definite integral.
Can it do ∫ 1/x dx?
Yes. ∫ 1/x dx = ln|x| + C, defined for x ≠ 0.
What does the Integral Calculator calculate?
Antiderivative of a function of x, with optional definite bounds. It returns Antiderivative and Definite integral.
What formula does the Integral Calculator use?
∫ f(x) dx = F(x) + C; ∫ₐᵇ f(x) dx = F(b) − F(a). The antiderivative is verified by differentiating it numerically and comparing with the original function.
What do I need to enter in the integral Calculator?
You enter Function of x. Optional: Lower bound (optional) and Upper bound (optional).
How is the result rounded or limited?
Symbolic result; numeric values to six decimals. Handles elementary functions of one variable x; anything the engine cannot simplify or confirm is reported as unsupported; antiderivatives of 1/x are shown as ln|x| + C in the usual convention.
What is an example of a integral result?
With function of x x^2*e^x, lower bound (optional) 0 and upper bound (optional) 1, the antiderivative is e^x*(2-2*x+x^2) + C.
Where does the method come from?
The method follows: Nerdamer symbolic engine, cross-checked numerically against the input; reference values from SymPy.
Related calculators
- Derivative Calculator — Derivative of a function of x, with a numeric check.
- Limit Calculator — Two-sided limit of a function of x, with a numeric check.
- Fraction Calculator — Add, subtract, multiply and divide fractions with simplified and mixed-number results.
- Ratio Calculator — Simplify a ratio of two whole numbers.
- Random Number Generator — Random integers in a range, optionally repeatable with a seed.
- Area Calculator — Area of common two-dimensional shapes.
Sources
- Nerdamer symbolic engine, cross-checked numerically against the input; reference values from SymPy
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.