Integral Calculator with Steps
Find supported indefinite integrals or evaluate proper definite integrals. Review the antiderivative, bounds, signed result, average value, numerical check and rule-based steps.
Last Updated: July 26, 2026Online Integral and Antiderivative Calculator
Enter f(x), choose a calculation mode and add finite bounds for a definite integral. Trigonometric functions use radians.
Integral result
∫ from 0 to 3 of x^2 dx = 9
The symbolic value and adaptive numerical check agree.
Integration steps
- Read f(x) = x^2 and identify a power of x.
- Apply the power rule: increase the exponent by 1 and divide by the new exponent.
- Use F(x) = 1/3·x^3 and calculate F(3) - F(0).
- Confirm the signed value 9 with adaptive numerical integration.
How to Use This Integral Calculator
- Enter a real-valued function of x. Use parentheses for every named function, such as sin(x), ln(x) or sqrt(x + 1).
- Choose Definite integral when you have lower and upper bounds. Choose Indefinite integral when you need an antiderivative F(x) + C.
- For a definite integral, enter finite constant bounds. The fields accept values and constant expressions such as -2, 1/3, pi and pi/2.
- Choose a signed net integral or total area. Signed mode allows positive and negative regions to cancel. Total-area mode integrates |f(x)| numerically.
- Select the display precision, choose Calculate Integral and review the formula, value, method, error estimate and steps.
The default example evaluates the integral of x2 from 0 to 3. The antiderivative is x3/3, so the Fundamental Theorem of Calculus gives 33/3 - 03/3 = 9. The function's average value on this interval is 9/(3 - 0) = 3.
What Is an Integral?
An integral describes accumulation. It combines many small contributions into one total. Depending on the problem, the accumulated quantity may represent signed area, distance, change in position, work, mass, probability or total cost.
Integration also reverses differentiation. If F'(x) = f(x), then F is an antiderivative of f. Because the derivative of every constant is zero, F(x) + C represents a whole family of antiderivatives. A definite integral adds bounds and returns one number instead of a family of functions.
The second identity applies when the hypotheses of the Fundamental Theorem of Calculus are satisfied. Domain restrictions still matter. A formula for an antiderivative does not permit an interval to cross a point where the original integrand is undefined.
Definite vs Indefinite Integrals
| Feature | Indefinite integral | Definite integral |
|---|---|---|
| Notation | ∫f(x) dx | ∫ from a to b of f(x) dx |
| Output | A family F(x) + C | One signed numerical value |
| Bounds | Not used | Required finite lower and upper values |
| Main idea | Reverse differentiation | Measure net accumulation on an interval |
| Calculator method | Supported symbolic patterns | Symbolic evaluation when available plus adaptive numerical checking |
Use indefinite mode when a problem asks you to integrate, find an antiderivative or solve a differential step. Use definite mode when the problem supplies an interval or asks for accumulated change. If you need geometric area, select Total area using |f(x)| rather than assuming a signed integral always equals area.
Supported Notation and Functions
Use x as the variable. The parser accepts numbers, decimals, scientific notation, +, -, *, /, ^, parentheses, pi, e and common implicit multiplication. Valid examples include 2x, 3(x + 1), 2sin(x) and (x + 1)(x - 1). Function names need parentheses.
The evaluator supports sin, cos, tan, sec, csc, cot, asin, acos, atan, sinh, cosh, tanh, sqrt, abs, ln, log and exp. The name log means base 10, while ln means the natural logarithm. Trigonometric inputs use radians.
Symbolic integration covers polynomials, sums, differences, constant multiples, powers of a linear expression, reciprocal linear expressions and many elementary functions with a linear inner expression. It also recognizes 1/(1 + x2) and 1/sqrt(1 - x2). Definite mode uses numerical integration for a wider set of continuous real-valued functions when no supported elementary antiderivative is found.
Core Integration Formulas
These formulas cover many introductory problems. In the table, c and n are constants, and u = ax + b has nonzero constant slope a.
| Integrand | Antiderivative | Condition |
|---|---|---|
| c | cx + C | c is constant |
| xn | xn+1/(n + 1) + C | n ≠ -1 |
| 1/x | ln|x| + C | x ≠ 0 |
| sin(u) | -cos(u)/a + C | u = ax + b |
| cos(u) | sin(u)/a + C | u = ax + b |
| exp(u) | exp(u)/a + C | u = ax + b |
| 1/(1 + x2) | atan(x) + C | All real x |
Integration is linear. You may integrate a sum term by term and move a constant multiplier outside the integral. Product and quotient rules do not reverse into simple product and quotient integration rules. More advanced products may require substitution, integration by parts, partial fractions or an identity outside this calculator's symbolic scope.
Worked Example: Indefinite Integral
Find ∫(3x2 - 4x + 5) dx. Apply linearity so each term is integrated separately. Increase each power of x by one, divide by the new exponent and retain the constant multiplier.
Combine the terms and include the constant of integration:
Differentiate the result to check it. The derivative is 3x2 - 4x + 5, which matches the original integrand.
Worked Example: Definite Integral
Evaluate ∫ from 0 to 2 of (3x2 + 2) dx. An antiderivative is F(x) = x3 + 2x. Substitute the upper and lower bounds in that order.
The definite integral is 12. The integrand is positive throughout [0, 2], so its signed integral and total geometric area are equal. Its average value is 12/(2 - 0) = 6.
Fundamental Theorem of Calculus
The Fundamental Theorem connects derivatives and definite integrals. When f is continuous on [a, b] and F is an antiderivative of f, the accumulated signed value equals F(b) - F(a). This converts a limit of sums into two function evaluations.
The order is important. Evaluate the upper bound first, then subtract the lower-bound value. Reversing the bounds reverses the sign:
Equal bounds give zero because F(a) - F(a) = 0. This zero-width result says there is no interval over which accumulation occurs.
Why Indefinite Integrals Need + C
If F'(x) = f(x), then [F(x) + 7]' and [F(x) - 20]' also equal f(x). Differentiation removes every constant. The symbol C records all possible vertical shifts of the antiderivative.
A definite integral does not need + C in its final value. If the same constant is included in both endpoint evaluations, it cancels: [F(b) + C] - [F(a) + C] = F(b) - F(a). The calculator still displays the symbolic antiderivative with + C so the underlying family remains clear.
Signed Integral vs Total Area
A definite integral is signed. Regions above the x-axis contribute positively. Regions below it contribute negatively. These contributions may cancel even when the graph encloses substantial geometric area.
Total-area mode numerically integrates |f(x)|. For example, the integral of x from -1 to 1 is zero because the negative and positive halves cancel. The total area is 1 because each triangular region has area 1/2.
Symbolic Results and Numerical Approximation
A symbolic result gives an antiderivative formula. It is useful for exact algebra, repeated evaluation and checking with differentiation. Not every elementary-looking integrand has an elementary antiderivative. For example, exp(-x2) cannot be expressed with the elementary functions supported here.
Definite mode therefore includes adaptive Gauss-Kronrod integration. The algorithm compares two nested quadrature estimates, subdivides intervals with the largest estimated error and continues until it reaches a controlled tolerance. It reports an approximation and an estimated numerical error. When a supported symbolic antiderivative exists, the calculator also evaluates F(b) - F(a) and compares it with the numerical result.
An error estimate is not a proof for every possible input. Rapid oscillation, sharp peaks and near-singular behavior can demand specialist methods. The calculator stops with a controlled message when its convergence and continuity checks do not support a reliable proper integral.
Domain Restrictions and Discontinuities
The definite-integral engine expects a proper real-valued integral on the full closed interval. It rejects an evaluated point when a denominator is zero, a logarithm has a nonpositive argument, a square root has a negative argument or a trigonometric function is undefined.
- 1/x cannot be integrated across x = 0 as an ordinary proper integral.
- ln(x) needs x > 0 throughout the interval.
- sqrt(x) needs x ≥ 0 throughout the interval.
- tan(x) and sec(x) are undefined where cos(x) = 0.
- cot(x) and csc(x) are undefined where sin(x) = 0.
- A fractional power of a negative base may have no supported real value.
Some improper integrals converge even though an endpoint or interior point is undefined. Others diverge or require a Cauchy principal value. This calculator does not label a principal value as an ordinary integral. Use a specialist improper-integral method when the interval includes a singularity or infinite bound.
Supported Input and Calculator Limits
Inputs longer than 180 characters, unsupported names, unmatched parentheses, excessive nesting and oversized expression trees receive controlled errors. Bounds must be finite constant expressions and cannot contain x. The tool does not accept infinity as a bound.
Indefinite mode does not cover every substitution, integration by parts, partial-fraction decomposition, trigonometric identity, special function, piecewise function or multiple integral. Definite mode handles a broader continuous numerical scope, but it does not support complex-valued paths, infinite intervals or intentionally singular principal-value problems.
The expression is parsed into a mathematical tree. It is never executed as JavaScript. The engine does not use eval, create executable functions from text or send the expression to a calculation server.
Common Integration Mistakes
- Forgetting to add C to an indefinite integral.
- Using the power rule with n = -1 instead of the logarithm rule for 1/x.
- Increasing an exponent without dividing by the new exponent.
- Forgetting the factor from a linear inner function such as 2x + 1.
- Subtracting F(b) from F(a) instead of calculating F(b) - F(a).
- Calling a signed integral total area when the function crosses below zero.
- Crossing an undefined point because the endpoint values appear finite.
- Rounding bounds or intermediate values before the final step.
Use the displayed method and steps to check your structure. For coursework, also show the algebra and notation required by your instructor.
Where Integrals Are Used
- Motion: Integrating velocity gives displacement, while integrating speed gives distance.
- Physics and engineering: Integrals calculate work, charge, mass, fluid force and accumulated energy.
- Economics: A marginal function can be integrated to recover a change in total cost or revenue.
- Probability: Integrating a continuous density over an interval gives probability.
- Geometry: Definite integrals measure areas, volumes and arc-related quantities using the correct setup.
- Data and models: Integration aggregates a changing rate across time, distance or another continuous variable.
The calculator evaluates the function you enter. It does not choose units, build a physical model or prove that a formula fits the real situation. Confirm the setup before relying on the number.
Accuracy, Privacy and Review Basis
Symbolic patterns use deterministic algebra rules. Numerical evaluation uses standard browser floating-point arithmetic and adaptive subdivision. The precision selector changes displayed digits. It does not create more information than the function and numerical method provide.
The definitions and Fundamental Theorem follow OpenStax Calculus Volume 1: The Definite Integral and The Fundamental Theorem of Calculus. Formula and substitution coverage follows the OpenStax sections on integration formulas and substitution. Numerical-method wording was checked against OpenStax Numerical Integration.
Your function and bounds stay in the current browser page. Reloading clears them. Report a repeatable issue through the 1Dollars contact page with the exact expression, mode, bounds, result type and precision used.
Integral Calculator vs Related Math Tools
| Tool | Best use | Main output |
|---|---|---|
| Integral Calculator | Find supported antiderivatives or evaluate proper definite integrals | Symbolic form, numerical value and steps |
| Graphing Calculator | Inspect where a function is positive, negative or discontinuous | Plot, trace and value table |
| Algebra Calculator | Simplify an expression before integrating | Algebraic result and steps |
| Scientific Calculator | Evaluate endpoint and constant expressions | Numerical result |
| Equation Calculator | Find roots that split signed-area intervals | Supported solutions |
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Frequently Asked Questions
What does an integral calculator find?
It finds a supported antiderivative F(x) + C or evaluates a proper definite integral between finite bounds. Definite mode also provides a numerical check and average value.
Does the integral calculator show steps?
Yes. It identifies supported polynomial, power, reciprocal, linear-substitution and elementary-function rules, then shows endpoint substitution or the numerical method used.
What is the difference between definite and indefinite integrals?
An indefinite integral is a family of antiderivatives and includes + C. A definite integral uses lower and upper bounds and returns one signed accumulated value.
Why does an indefinite integral include + C?
Every constant has derivative zero. Adding any constant to an antiderivative produces another valid antiderivative, so + C represents the full family.
Does the calculator use degrees or radians?
It uses radians. Standard calculus formulas for trigonometric derivatives and integrals assume radian measure.
Can it calculate total area under a curve?
Yes. Choose Total area using |f(x)|. Signed mode may cancel regions below and above the x-axis, while total-area mode makes both contributions positive.
What happens when the bounds are reversed?
The value changes sign. Integrating from b to a equals the negative of integrating from a to b. Total-area mode keeps the orientation sign while measuring |f(x)|.
Why was my definite integral rejected?
The interval may contain an undefined point, singularity, nonreal value or behavior that did not pass the numerical convergence checks. This tool evaluates proper real-valued integrals.
Can it solve every indefinite integral?
No. It covers documented elementary patterns. Advanced substitution, integration by parts, partial fractions, piecewise forms and special functions may require another method.
Are entered functions and bounds stored?
No calculation server receives them. Parsing, symbolic integration and numerical evaluation run in the current browser page.
Educational and financial disclaimer: This calculator provides general mathematical results from your inputs. It is not financial, scientific, engineering, tax or professional advice. Verify functions, domains, bounds, units, assumptions and required precision before using a result for an important decision.