Examples for

Integrals

Integrals come in two varieties: indefinite and definite. Indefinite integrals can be thought of as antiderivatives, and definite integrals give signed area or volume under a curve, surface or solid. Wolfram|Alpha can compute indefinite and definite integrals of one or more variables, and can be used to explore plots, solutions and alternate representations of a wide variety of integrals.
Indefinite Integrals

Find the antiderivatives of mathematical expressions.

Compute an indefinite integral:

int x^5 dx
integrate x^2 sin^3 x dx
int e^t sin(5t) dt

Compute an indefinite integral that cannot be expressed in elementary terms:

int e^(-t^2) dt
integrate 1/sqrt(1-u^4)

Generate a table of integrals that contain a given function:

integrals containing cos(u)
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Definite Integrals

Find integrals with lower and upper limits, also known as Riemann integrals.

Compute a definite integral:

integrate sin x dx from x=0 to pi
int e^(-a t) dt, t=0..a

Compute an improper integral:

int sinx/x dx, x=0..infinity

Generate a table of definite integral formulas:

definite integrals containing exp(t)
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Multiple Integrals

Calculate definite nested integrals in multiple variables.

Compute a multiple integral:

integrate x^2 sin y dx dy, x=0..1, y=0..pi
int (x^2 y^2 + x y^3) dx dy, x=-2 to 2, y=-2 to 2
int sin^2 x + y sin z dx dy dz , x=0..pi, y=0..1, z=0..pi

Compute an integral over an unbounded region:

int e^-(x^2+y^2) dx dy, x=-oo to oo, y=-oo to oo
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Numerical Integration

Integrate expressions using numerical approximation.

Numerically integrate functions that cannot be integrated symbolically:

integrate sin(cos x) from x=0 to 1

Approximate an integral using a specified numerical method:

5 interval trapezoidal rule integrate sinx cosx on [0,4]
integral (x^2-2)/x dx from 1 to 2 using Boole's rule
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Integral Representations

Explore integral representations of various math functions.

Find integral representations for a function:

log x integral representation
gamma(x) integral representation
Fresnel S(x) integral rep
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Integrals Related to Special Functions

Find definite or indefinite integrals involving a particular special function.

Explore interesting indefinite integrals containing special functions:

sample integrals besselj
integrals using the airy function
integrals with li(x)

Explore interesting definite integrals containing special functions:

sample definite integrals product log
definite integrals with LegendreP
definite integrals with Si(x)
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