Difference between revisions of "Indefinite integral"
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An '''indefinite integral''', or antiderivative, is an [[integral]] without upper and lower limits. | An '''indefinite integral''', or antiderivative, is an [[integral]] without upper and lower limits. | ||
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=== Polynomial and simple rational === | === Polynomial and simple rational === | ||
| − | To see the proofs for the first two integrals, see | + | To see the proofs for the first two integrals, see Riemann integral. |
:<math>\int xdx = {1 \over 2} x^2 + C</math> | :<math>\int xdx = {1 \over 2} x^2 + C</math> | ||
:<math>\int x^2dx = {1 \over 3} x^3 + C</math> | :<math>\int x^2dx = {1 \over 3} x^3 + C</math> | ||
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:<math>\int \cos(x)dx = \sin(x) + C</math> | :<math>\int \cos(x)dx = \sin(x) + C</math> | ||
:<math>\int \tan(x)dx = \ln|\sec(x)| + C</math> | :<math>\int \tan(x)dx = \ln|\sec(x)| + C</math> | ||
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=== Exponential === | === Exponential === | ||
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*[[Definite integral]] | *[[Definite integral]] | ||
| − | + | [[Category:Integration]] | |
| − | [[ | + | [[Category:Calculus]] |
Latest revision as of 05:09, February 10, 2017
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This article/section deals with mathematical concepts appropriate for late high school or early college. |
An indefinite integral, or antiderivative, is an integral without upper and lower limits.
Contents
Indefinite Integrals
There are an infinite number of antiderivatives for a given function, because each indefinite integral can have an arbitrary constant added to it which disappears upon differentiation. However, the fundamental theorem of calculus relates a definite integral to an indefinite integral by taking its value at the boundary points.
Whenever any expression is integrated the constant of integration, 
, is always added.
A list of simple antiderivatives
The identity antiderivative:
Polynomial and simple rational
To see the proofs for the first two integrals, see Riemann integral.
The general rule for polynomial expressions is:
Note: 
. See below for when 
Rational
For a more detailed treatment, see Partial fractions in integration.
Rational antiderivatives are much more difficult and follow different rules.
