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Calculator Answers

Integral Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

An integral calculator is a calculus utility designed to compute the definite integral of a mathematical function over a specified interval. For polynomial functions, the calculator applies the power rule of integration analytically. For more complex or non-polynomial expressions, it utilizes numerical integration techniques, such as Simpson's rule, to approximate the area under the curve. Students, engineers, and physics researchers use these calculators to verify integration homework, solve area and volume problems, and evaluate physical systems represented by continuous functions.

Type a polynomial in x and we'll integrate it term by term using the power rule. Choose indefinite for the antiderivative or definite to evaluate between two bounds — with a numeric Simpson's rule cross-check.

See also: Area Between Two Curves Calculator, Derivative Calculator, and Double Integral Calculator.

Quick Answer

Compute the definite integral of a function. Enter the mathematical expression, variable of integration, and the lower and upper bounds.

Polynomials in x only — terms like 3x^2, -2x, and 5. No functions like sin or log.

Mode

Antiderivative

∫ f(x) dx =

x^3 +x^2 +x + C

Power rule applied term by term, plus constant of integration

f(x) (parsed)3x^2 +2x +1
Number of terms in F3

Polynomial integrals only — power rule, term by term. The numeric row uses Simpson's rule on 1,000 subintervals as a sanity check. For trig, exponential, logarithmic, or composite integrands, use a full CAS.

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Examples

∫ 3x² + 2x + 1 dx

x³ + x² + x + C

∫ x² dx, from 0 to 2

8/3 ≈ 2.667

∫ 4x dx, from 1 to 3

16

∫ 6x² dx, from −1 to 1

4

How it works

Each polynomial term integrates with the power rule. Add a constant of integration C for indefinite integrals, or evaluate the antiderivative at both endpoints for definite integrals.

power rule · ∫ c·xⁿ dx = c·x^(n+1) / (n+1) + C

definite · ∫ from a to b of f(x) dx = F(b) − F(a)

For a sanity check, the definite mode also runs Simpson's rule with 1,000 subintervals — for polynomials the two should match.

Supported: polynomials in x · ^ for exponents · constants · negative coefficients. Unsupported: 1/x, sin, cos, ln, e^x, products, quotients, substitution.

What is an integral calculator?

An integral calculator computes antiderivatives (indefinite integrals) and evaluates areas under curves (definite integrals). While advanced calculus integrates a wide variety of logarithmic and trigonometric functions, this calculator is specifically optimized to step through polynomial functions term by term, displaying precise fractions and numerical approximations.

How to integrate polynomials (The Power Rule)

Integration is the reverse process of differentiation. For a polynomial term, you apply the Power Rule of integration:

  1. Increase the variable's exponent by 1 (e.g., x² becomes x³).
  2. Divide the coefficient by this new exponent value.
  3. For constants (e.g., 5), multiply by the variable (e.g., 5x).
  4. If the integral is indefinite, add the integration constant + C.

Definite integrals and the Fundamental Theorem of Calculus

To calculate a definite integral with an upper bound b and a lower bound a, find the antiderivative function F(x) first. Then, evaluate F(b) and subtract F(a). The constant C cancels out during subtraction, leaving an exact numerical value representing the area under the curve.

Worked example: Evaluating a definite integral

Let's evaluate the integral of 3x² + 4x from 1 to 3:

  • Integrand: f(x) = 3x² + 4x
  • Find antiderivative: F(x) = (3x³ ÷ 3) + (4x² ÷ 2) = x³ + 2x²
  • Evaluate at upper bound (3): F(3) = 3³ + 2(3²) = 27 + 18 = 45
  • Evaluate at lower bound (1): F(1) = 1³ + 2(1²) = 1 + 2 = 3
  • Subtract limits: F(3) − F(1) = 45 − 3 = 42

The definite integral value is exactly 42.

Common mistakes when calculating integrals

  • Forgetting the constant of integration (+ C): Omitting `+ C` on indefinite integrals. This is a common deduction on math exams.
  • Incorrect subtraction order: Evaluating the lower bound first and subtracting the upper bound (computing F(a) − F(b) instead of F(b) − F(a)).
  • Misapplying the power rule to x⁻¹: Attempting to integrate 1/x (or x⁻¹) by writing x⁰/0, which causes division by zero. The correct antiderivative of 1/x is ln|x|.
  • Arithmetic sign errors: Making calculation mistakes when plugging negative bounds into the evaluated antiderivative.
Polynomials only. For 1/x, trig, exponential, logarithmic, or composite integrands, use a full computer-algebra system. We deliberately keep the scope narrow so every step is auditable.

For the limit definition behind Riemann sums, see the limit calculator.

Related Calculators

More tools from Education

Area Between Two Curves CalculatorCompute the area enclosed between two functions f(x) and g(x) over a given interval using definite integration.Derivative CalculatorDifferentiate polynomials term by term with the power rule — clear breakdown of each term's derivative.Double Integral CalculatorCalculate the definite double integral of a function of two variables over rectangular regions using numerical integration.Partial Fraction Decomposition CalculatorDecompose (ax + b) / ((x − r₁)(x − r₂)) into A / (x − r₁) + B / (x − r₂) for distinct linear factors. Returns A, B, and the decomposed form with step notes.

Frequently asked questions

Polynomials in x only — terms like 3x^2, -2x, and constants. The calculator does not handle 1/x, trig, exponential, logarithmic, or composite integrands. For anything beyond polynomial integration, use a full computer algebra system like Wolfram Alpha or SymPy.

For any term c·xⁿ where n ≠ −1, the antiderivative is c·x^(n+1) / (n+1). We apply this to each term and add the results. The constant of integration C captures the family of all functions whose derivative equals f(x).

By the Fundamental Theorem of Calculus: ∫ from a to b of f(x) dx = F(b) − F(a), where F is any antiderivative of f. We compute F using the power rule, evaluate at both endpoints, and subtract. We also run Simpson's rule numerically (n = 1,000 subintervals) and report it alongside as a sanity check — for polynomials of degree ≤ 3 the two should match exactly to floating-point precision.

The power rule fails when n = −1: dividing by (n + 1) means dividing by zero. The correct antiderivative is ln|x|, which falls outside the polynomial scope of this calculator. We flag the case rather than print a wrong answer.

An indefinite integral represents the family of all antiderivatives of a function and always includes a constant of integration (+ C). A definite integral calculates the net signed area under a curve between two numerical bounds, resulting in a single real number.

When you differentiate a constant, the result is zero. Because of this, when finding an indefinite integral (antiderivative), there could have been any constant number in the original function. The '+ C' represents this unknown constant value.

Simpson's rule is highly accurate for smooth functions and yields exact results for polynomials of degree 3 or lower. For higher-degree polynomials, it provides an extremely close numerical approximation.

Integration is used in physics, engineering, and economics to calculate areas under curves, volumes of solids, work done by variable forces, total accumulation of quantities over time, and consumer/producer surplus.