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Integral Calculator

Read asf(x)=x2f(x) = x^{2}

Powers ^ · multiply with * or a space · sqrt( ) ln( ) log( ) sin( ) cos( ) tan( ) e pi |x| · implicit products like 2x or x sin x work

Try an example

Numbers, pi, e or -inf

Leave both bounds empty for the antiderivative only

Definite integral

∫03x2 dx=[x33]03=9−0=9\int_{0}^{3} x^{2}\,dx = \left[\frac{x^{3}}{3}\right]_{0}^{3} = 9 - 0 = 9

Average value of f on the interval: 3

Indefinite integral (antiderivative)

∫x2 dx=x33+C\int x^{2}\,dx = \frac{x^{3}}{3} + C
x^3/3 + C

Checked by differentiating the result back to f(x).

Graph

-0.50.00.51.01.52.02.53.03.5024681012
f(x)area above axisbelow axis

Methods used

  • Power rule∫xn dx=xn+1n+1+C(n≠−1)\int x^{n}\,dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)

Find antiderivatives and definite integrals of functions of x. Definite integrals accept infinite bounds and are computed numerically when no elementary antiderivative exists; each antiderivative is checked by differentiating it back to the original function.

Formula

∫xndx=xn+1/(n+1)+C(n≠−1)\int \text{x}^{\text{n}} \text{dx} = \text{x}^{\text{n}+1}/(\text{n} + 1) + \text{C} (\text{n} \ne - 1)
∫abf(x)dx=F(b)−F(a)\int _{\text{a}}^{\text{b}} \text{f}(\text{x}) \text{dx} = \text{F}(\text{b}) - \text{F}(\text{a})

Worked Example

f(x)x²
From0
To3
Result
Definite integral9
Antiderivativex³/3 + C
Average value on [0, 3]3

An antiderivative of x² is x³/3, so the integral from 0 to 3 is 27/3 − 0 = 9: the area under the parabola over that interval. Dividing by the width of the interval, 3, gives the average value of the function.

Frequently Asked Questions

What is the difference between a definite and an indefinite integral?

An indefinite integral is a family of functions, F(x) + C, whose derivative is f(x). A definite integral is a single number, the signed area between f and the x-axis from a to b, equal to F(b) − F(a) by the fundamental theorem of calculus.

Why do some functions have no antiderivative formula?

Every continuous function has an antiderivative, but for some, such as e^(−x²), sin(x)/x and √(1 + x³), it can't be written with elementary functions. Their definite integrals can still be computed numerically: e^(−x²) integrated over the whole real line gives √π ≈ 1.7725.

What does a negative definite integral mean?

More of the area lies below the x-axis than above it on that interval. The integral of sin x from 0 to 2π is 0 because the two halves cancel; integrate |f(x)| to get the total area instead.

Common Mistakes

  • !Leaving out the constant of integration, + C, on an indefinite integral.
  • !Using ∫ xⁿ dx = xⁿ⁺¹/(n + 1) for n = −1: the integral of 1/x is ln|x| + C.

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