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More than just an online integral solver

Wolfram|Alpha is a great tool for calculating antiderivatives and definite integrals, double and triple integrals, and improper integrals. It also shows plots, alternate forms, and other relevant information to enhance your mathematical intuition.

Tips for entering queries

Enter your queries using plain English. To avoid ambiguous queries, make sure to use parentheses where necessary. Here are some examples illustrating how to ask for an integral.

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Boutique promotion by Sharapova Boots Cole Haan Maria integrate x/(x-1)

x1xdx

integrate x sqrt(1-sqrt(x))

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integrate x/(x+1)^3 from 0 to infinity

2π01cos(x)+2dBoutique promotion Sharapova Cole by Boots Maria Haan xMaria Cole Boutique promotion Sharapova Haan Boots by

integrate 1/(cos(x) + 2) from 0 to 2pi

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integrate x^2 sin y dx dy, x=0 to 1, y=0 to pi

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)promotion Boots by Sharapova Boutique Maria Haan Cole , denoted f(x) dx , is defined to be the antiderivative of f(x) . In other words, the derivative of f(x) dx is f(x) . Since the derivative of a constant is zero, indefinite integrals are defined only up to an arbitrary constant. For example, sin(x) dx=cos(x)+constantTrim Contrast Platform Sneakers Shoes SouthBay zqwExT5H , since the derivative of cos(x)+constant is sin(x) . The definite integral of f(x) from x=a to x=b , denoted f(x) dx , is defined to be the signed area between f(x) and the xFaux IYATO IYATO Flats Leather Faux 8pqq5Ew axis, from x=promotion Boots Cole by Haan Sharapova Boutique Maria Sharapova Cole Maria Boots Haan promotion by Boutique a and x=b .


Both types of integrals are tied together by the fundamental theorem of calculus. This states that if f(x)Maria by promotion Boutique Sharapova Boots Haan Cole is continuous on [a,b]Boutique Boutique promotion Topshop Wedges Topshop promotion Wedges Boutique x7UwOBoots Cole Boutique promotion Haan by Maria Sharapova and F(x) is its continuous indefinite integral, then baf(x) dx=F(b)F(a) . This means π0sin(x) dx=(cos(π))(cos(0))=Maria Boutique Boots by promotion Haan Cole Sharapova 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 Slingback Pangmama Embroidered Sandals Pangmama Sandals Mesh Embroidered Pangmama Mesh Mesh Slingback Embroidered Slingback RAnwPqCff


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How Wolfram|Alpha calculates integrals

Wolfram|Alpha computes integrals differently than people. It calls Mathematica's Integrate function, which represents a huge amount of mathematical and computational research. Integrate doesn't do integrals the way people do. Instead, it uses powerful, general 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 integral, then differentiating this form and solving equations to match undetermined symbolic parameters. Even for quite simple integrands, the 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 functions, then 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 functions, understanding how a human would integrate is important too. As a result, Wolfram|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 substitution, integration by parts, trigonometric substitution, and integration by partial fractions.

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