Integral Calculator

Integration is the reverse of differentiation, and a definite integral gives the signed area under a curve. This integral calculator finds antiderivatives using standard integrals, linear substitution and integration by parts, then evaluates definite integrals between limits. It shades the area and checks every answer numerically.

Leave the limits blank for an indefinite integral.
Try:

How to use the integral calculator

  1. Type the function to integrate, such as x^2 or x e^x.
  2. Enter lower and upper limits for a definite integral, or leave them blank.
  3. Press Calculate to see the integral, the method and the area graph.

Formula

∫xn dx=xn+1n+1+C\int x^n\,dx = \frac{x^{n+1}}{n+1} + C
∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b) - F(a)

Worked example: ∫₀³ x² dx

  • f(x) =: x^2
  • Lower limit (optional): 0
  • Upper limit (optional): 3

✓ Answer checked

∫03x2 dx=9\int_{0}^{3} x^{2}\,dx = 9
−112345101520f(x)
Shaded: the signed area under f from 0 to 3. The 12 rectangles (midpoint rule) approach it as they get thinner.
Step-by-step working (3 steps)

  1. Read the function

    This is how the input was understood.

    f(x)=x2f(x) = x^{2}

  2. Power rule for each term ∫xⁿ dx = xⁿ⁺¹/(n+1)

    Expand into powers of x, then raise each power by one and divide by the new power.

    ∫x2 dx=13x3\int x^{2}\,dx = \frac{1}{3}x^{3}

  3. Evaluate between the limits Fundamental theorem of calculus

    Substitute the upper limit, then subtract the value at the lower limit.

    [13x3]03=9−0=9\left[\frac{1}{3}x^{3}\right]_{0}^{3} = 9 - 0 = 9
Formulas used
∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b) - F(a)
How this was checked
  • ✓ Numerical integration (adaptive Simpson) gives 9, matching F(b) − F(a).

Frequently asked questions

What is the + C in an indefinite integral?

Every constant has derivative 0, so there are infinitely many antiderivatives that differ by a constant. The + C stands for all of them.

What does a definite integral mean?

It is the signed area between the curve and the x-axis from a to b. Area below the axis counts as negative.

What if the calculator finds no formula?

For a definite integral it gives a numerical value, clearly labelled as approximate. For an indefinite integral it says no closed form was found.

Related calculators

Derivative Calculator

Differentiate any function with every rule shown: power, product, quotient and chain rules.

Area Calculator

Find the area of a rectangle, square, triangle, circle, trapezoid or parallelogram.

Graphing Calculator

Plot up to three functions with an animated graph.

Limit Calculator

Evaluate limits step by step: direct substitution, L'Hôpital's rule for 0/0, and limits at infinity.

Related guides