What is an integral calculator?
An indefinite integral describes a family of antiderivatives F(x) + C whose derivative is the entered function. A definite integral measures signed accumulation between two bounds. This calculator provides symbolic antiderivatives for supported forms and a numerical definite-integral approximation for supported continuous real expressions.
Use the Derivative Calculator to differentiate an antiderivative as a separate check. The Graphing Calculator helps visualize where an integrand lies above or below the horizontal axis. The definite integral adds these contributions with their signs; it is not automatically the total geometric area.
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Optional calculus workbooks (paid link) can help you practice power-rule integration, substitution and integration by parts before checking your own working. The online calculator is free to use without buying anything.
How to find an indefinite or definite integral
Enter the integrand and select x, y or t as the variable. Choose Indefinite integral for a symbolic answer plus C. Choose Definite integral to enable lower and upper bounds. Enter finite constant bounds such as 0 and pi. Select Calculate; edit any field to clear stale results, or Reset to clear the form. Result actions include the method and steps.
Start with a supported example such as x^2, sin(2x) or 2x*cos(x^2). Use parentheses around a denominator or a complete exponent. A graph-paper notebook (paid link) is optional for recording bounds and sketching signed contributions. For coefficient checks, the Fraction Calculator can keep rational arithmetic separate from the calculus.
Expression syntax and angle conventions
Use your chosen variable (x, y or t), numbers, pi, e, parentheses and +, -, *, / or ^. Multiplication may be explicit, as in 2*x, or implicit, as in 2x or (x+1)(x-1). Write sin(x), cos(x), tan(x), exp(x), ln(x), sqrt(x), abs(x), asin(x), acos(x) or atan(x). Functions need parentheses. log(x) means the natural logarithm, just like ln(x). All trigonometric angles are in radians; inverse trigonometric answers are also in radians.
Powers associate to the right: x^2^3 means x^(2^3). A leading minus has lower precedence than a power, so -x^2 means -(x^2); use (-x)^2 to square the negative input. Only one selected variable is allowed. Bounds and evaluation points may be constant expressions such as pi/2, sqrt(2) or 1/3.
Integration rules and the constant of integration
The polynomial power rule integrates c*x^n to c*x^(n+1)/(n+1) for n other than -1. The reciprocal rule integrates 1/x to ln(abs(x)) on an interval that excludes zero. Sums, differences and constant factors are handled term by term. Linear-inner forms of sin, cos, tan, exp, ln, sqrt, abs and constant powers are supported.
Substitution recognizes a supported outer function multiplied by a constant multiple of its inner derivative, such as 2x*cos(x^2). A logarithmic substitution handles a numerator proportional to the denominator’s derivative. Polynomial factors of degree at most 8 multiplied by sin, cos or exp of a linear expression also support repeated integration by parts. Other symbolic forms may be unavailable even when a mathematical antiderivative exists.
Every indefinite answer includes + C. Different constants may apply on disconnected parts of the domain. For a continuous function with an antiderivative, the fundamental theorem gives the definite integral as F(b) - F(a). This page also uses adaptive Simpson quadrature for the numerical value and checks agreement with an available antiderivative.
Worked integral examples
Indefinite power rule: Enter x^2 and choose Indefinite integral. Increasing the exponent to 3 and dividing by 3 gives x^3/3 + C. Differentiating this expression returns x^2.
Definite integral: Enter x^2 with lower bound 0 and upper bound 2. F(2) - F(0) = 8/3, approximately 2.6666666667. The displayed numerical value is marked with ≈. Optional scientific calculators (paid link) can help check ordinary arithmetic and any integral mode supported by the chosen model.
Substitution: For 2x*cos(x^2), set u = x^2 so du = 2x dx. Integrating cos(u) gives sin(u), hence sin(x^2) + C. For exp(-x^2), symbolic integration is outside this version’s supported forms, but finite continuous bounds can be evaluated numerically.
Signed area, reversed bounds and numerical error
The definite integral is signed accumulation. For sin(x) from 0 to 2*pi, positive and negative contributions cancel to approximately zero; the total geometric area is different. Reversing the bounds reverses the sign. Equal finite bounds produce zero only when the integrand is defined at that point in this implementation.
The facts identify the numerical method, the bounds, the evaluation count and an estimated quadrature error. Adaptive Simpson integration starts with 128 subintervals, then refines them toward a combined absolute/relative error target. The estimate is useful for checking convergence but is not a rigorous mathematical bound. Optional graphing calculators (paid link) can help explore the same function; confirm whether the model computes signed numerical integrals and use matching bounds.
Supported integrals and real-domain limitations
This is a single-variable real calculator. It does not evaluate improper integrals, infinite bounds, principal values, multivariable integrals or discontinuities. For example, 1/x from -1 to 1 is rejected, even though symmetric sampling might appear to cancel. Endpoints that make ln, a denominator or another expression undefined are also rejected. A formula that simplifies algebraically may still have a hole in its original domain.
Conservative interval checks cover every initial subinterval before quadrature. They may reject a valid expression when its continuity cannot be established, particularly with dependent interval terms, sharp features or fast oscillation. Numerical quadrature can miss sufficiently narrow features or suffer cancellation; the estimated error does not guarantee accuracy for every expression. Use an independent analytic check for sensitive work.
Indefinite integration is rule based rather than a general computer algebra system. It supports rational-coefficient polynomials up to the engine’s degree limit, the forms above, and selected substitutions. Unsupported symbolic inputs show a clear message and may still work in definite mode. All angles are radians; log means ln. Noninteger powers of negative bases are outside the numerical convention.
The same 240-character, 160-token, 24-level nesting and rational-coefficient work limits apply. Bounds must be finite and between -1,000,000 and 1,000,000. Quadrature targets approximately 1e-8 absolute plus 1e-9 relative accuracy internally, has a limited evaluation budget and rejects nonconvergence. The displayed decimal is rounded to roughly 11 significant digits.
Integral Calculator FAQs
Sources and methodology
These educational references explain the underlying calculus. This independently implemented browser tool is not affiliated with their publisher. The supported forms, numerical method and limits are described above.