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Wolfram|Alpha is a great tool for calculating antiderivatives and definite integralsdouble and triple integralsand improper integrals. The Wolfram|Alpha Integral Calculator also shows plotsalternate forms and other relevant information to enhance your mathematical intuition.

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Use Math Input above or enter your integral calculator queries using plain English. To avoid ambiguous queriesmake sure to use parentheses where necessary. Here are some examples illustrating how to ask for an integral using plain English.

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What are integrals?

Integration is an important tool in calculus that can give an antiderivative or represent area under a curve.

The indefinite integral of f (x)f xdenoted Start IntegralStart integrand f (x) integrand End,Start first variable x first variable End Integral Endf xdxis defined to be the antiderivative of f (x)f x. In other wordsthe derivative of Start IntegralStart integrand f (x) integrand End,Start first variable x first variable End Integral Endf xdx is f (x)f x. Since the derivative of a constant is 0indefinite integrals are defined only up to an arbitrary constant. For example, Start IntegralStart integrand Start SineStart angle x angle End Sine End integrand End,Start first variable x first variable End Integral End = - Start CosineStart angle x angle End Cosine End +constantsinxdx = -cosx+constantsince the derivative of − Start CosineStart angle x angle End Cosine End + constant−cosx + constant is Start SineStart angle x angle End Sine Endsinx. The definite integral of f (x)f x from x = ax = a to x = bx = bdenoted Start Definite IntegralStart first lower limit a first lower limit End,Start first upper limit b first upper limit End,Start integrand f (x) integrand End,Start first variable x first variable End Definite Integral Endbaf xdxis defined to be the signed area between f (x)f x and the xx axisfrom x = ax = a to x = bx = b.

Both types of integrals are tied together by the fundamental theorem of calculus. This states that if f (x)f x is continuous on [a,b]a,b and F(x)Fx is its continuous indefinite integralthen Start Definite IntegralStart first lower limit a first lower limit End,Start first upper limit b first upper limit End,Start integrand f (x) integrand End,Start first variable x first variable End Definite Integral End = F(b) - F(a)baf xdx = Fb - Fa. This means Start Definite IntegralStart first lower limit 0 first lower limit End,Start first upper limit π first upper limit End,Start integrand Start SineStart angle x angle End Sine End integrand End,Start first variable x first variable End Definite Integral End = (- Start CosineStart angle π angle End Cosine End ) - (- Start CosineStart angle 0 angle End Cosine End ) = 2π0sinxdx = -cosπ - -cos0 = 2. Sometimes an approximation to a definite integral is desired. A common way to do so is to place thin rectangles under the curve and add the signed areas together. Wolfram|Alpha can solve a broad range of integrals ..

How Wolfram|Alpha calculates integrals

Wolfram|Alpha computes integrals differently than people. It calls Mathematica's Integrate functionwhich represents a huge amount of mathematical and computational research. Integrate does not do integrals the way people do. Insteadit uses powerfulgeneral algorithms that often involve very sophisticated math. There are a couple of approaches that it most commonly takes. One involves working out the general form for an integralthen differentiating this form and solving equations to match undetermined symbolic parameters. Even for quite simple integrandsthe equations generated in this way can be highly complex and require Mathematica's strong algebraic computation capabilities to solve. Another approach that Mathematica uses in working out integrals is to convert them to generalized hypergeometric functionsthen use collections of relations about these highly general mathematical functions.

While these powerful algorithms give Wolfram|Alpha the ability to compute integrals very quickly and handle a wide array of special functionsunderstanding how a human would integrate is important too. As a resultWolfram|Alpha also has algorithms to perform integrations step by step. These use completely different integration techniques that mimic the way humans would approach an integral. This includes integration by substitutionintegration by partstrigonometric substitution and integration by partial fractions.