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Integral Calculator

Find the indefinite integral (antiderivative) of a function of x with the rules used, and the definite integral between two limits, with a numeric check.

Results

ItemValue

Steps

StepWorking
Type x as the variable, with ^ for powers and brackets for function inputs, such as cos(3x). The calculator handles polynomials, powers of linear terms, exponentials, logarithms, trig and inverse trig of linear terms, and simple integration by parts. Each answer is checked by differentiating it. When no simple formula is found, the definite integral is still estimated numerically.

Quick Answer

An integral adds up a function. The indefinite integral, or antiderivative, of x^2 + 3x is x^3/3 + 3x^2/2 + C, using the power rule ∫xⁿ dx = xⁿ⁺¹/(n + 1). The definite integral from 0 to 2 is 8/3 + 6 = 8.666667.

Last updated: 2 November 2026 ·How we calculate this: standard integration rules, checked by differentiation

Table of Contents

How This Integral Calculator Works

Type a function of x and, if you want a number, the lower and upper limits. The calculator applies integration rules such as the power rule, the constant multiple and sum rules, linear substitution and integration by parts, then simplifies the answer. It checks the result by taking its derivative. With limits, it gives the definite integral, plus a numeric estimate using Simpson's rule.

Quick explanation: integration is the reverse of differentiation.

  • Indefinite integral: a family of functions, written F(x) + C.
  • Definite integral: a number, the signed area under the curve between two limits.
  • C: the constant of integration, because constants vanish when you differentiate.

How to Use the Integral Calculator

  1. Type your function using x as the variable.
  2. For a definite integral, enter the lower and upper limits. Leave both empty for only the antiderivative.
  3. Press Calculate.
  4. Read the antiderivative and the rules used.
  5. Compare the definite integral with the numeric estimate.

Core Integration Rules

  • Power rule: ∫xn dx = xn+1 / (n + 1) + C, for n ≠ −1
  • Reciprocal: ∫1/x dx = ln|x| + C
  • Constant multiple: ∫k·f(x) dx = k·∫f(x) dx
  • Sum rule: ∫(f + g) dx = ∫f dx + ∫g dx
  • By parts: ∫u dv = u·v − ∫v du
  • Definite integral: ∫ from a to b of f(x) dx = F(b) − F(a)

Key Terms

  • Integral: the result of adding up a function, as an antiderivative or an area.
  • Antiderivative: a function F whose derivative is f.
  • Constant of integration: the + C in an indefinite integral.
  • Limits of integration: the start and end values a and b in a definite integral.
  • Simpson's rule: a numeric method that estimates an integral using parabolas.

Worked Example

Find ∫x·ex dx using integration by parts.

  • Let u = x and dv = ex dx, so du = dx and v = ex.
  • ∫x·ex dx = x·ex − ∫ex dx
  • = x·ex − ex + C

Real-Life Examples

  • Area under a curve: ∫ from 0 to 3 of x2 dx = 9.
  • Distance from speed: if speed is v(t) = 4t, the distance from t = 0 to 5 is ∫4t dt = 50. Type t as x.
  • Average value: divide the definite integral by (b − a).
  • Homework checks: compare your antiderivative with the result, keeping in mind the + C.

Definite vs Indefinite Integral

  • Indefinite: gives a function, F(x) + C.
  • Definite: gives a number, F(b) − F(a).
  • Link: the fundamental theorem of calculus connects them.

Common Integrals

FunctionIntegral
k (a constant)kx + C
xn, n ≠ −1xn+1 / (n + 1) + C
1 / xln|x| + C
exex + C
axax / ln(a) + C
ln(x)x·ln(x) − x + C
sin(x)−cos(x) + C
cos(x)sin(x) + C
sec(x)2tan(x) + C
tan(x)−ln|cos(x)| + C
1 / (1 + x2)arctan(x) + C
1 / √(1 − x2)arcsin(x) + C

Tips for Typing Functions

  • Powers: type x^3, or x^(-2) for 1/x².
  • Roots: use sqrt(x) or x^(1/2).
  • Brackets: write cos(3x), not cos 3x.
  • Expand products: for (x + 1)(x + 2), enter x^2 + 3x + 2 for the best result.

Common Mistakes to Avoid

  • Forgetting + C. Every indefinite integral needs the constant of integration.
  • Using the power rule for 1/x. ∫x−1 dx is ln|x|, not x0/0.
  • Ignoring the inner slope. ∫cos(3x) dx = sin(3x)/3, not sin(3x).
  • Integrating across a break. 1/x is undefined at 0, so ∫ from −1 to 1 of 1/x dx does not exist.
  • Multiplying integrals for a product. ∫f·g dx is not ∫f dx · ∫g dx.

Frequently Asked Questions

A way to add up a function. It gives an antiderivative or the area under a curve.

x^3/3 + C, by the power rule.

ln|x| + C.

e^x + C.

−cos(x) + C. The integral of cos(x) is sin(x) + C.

The derivative of a constant is zero, so any constant could be part of the antiderivative.

Find the antiderivative F(x), then work out F(b) − F(a).

Yes. Area below the x-axis counts as negative.

Some functions, like e^(x^2), have no simple antiderivative, and this tool handles common forms only. The definite integral is still estimated.

Yes. Trig functions use radians.

A rule for products: ∫u dv = u·v − ∫v du. It works well for x·e^x or x·sin(x).

Very accurate for smooth functions, usually to six or more digits. It may fail where the function is undefined.

References & Sources

SourceUsed ForLink
OpenStax, Calculus Volume 1, Chapter 5: Integration Free textbook on antiderivatives, definite integrals and the fundamental theorem. https://openstax.org/books/calculus-volume-1/pages/5-introduction
Wikipedia, Lists of integrals Tables of standard antiderivatives. https://en.wikipedia.org/wiki/Lists_of_integrals