Difference between revisions of "Indefinite integral"
m (→Polynomial and simple rational: added a link to Riemann Integral) |
m (→Exponential: fixed error) |
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=== Exponential === | === Exponential === | ||
:<math>\int e^x = e^x + C</math> | :<math>\int e^x = e^x + C</math> | ||
| − | :<math>\int a^x = \ln(a) | + | :<math>\int a^x = {a^x \over \ln(a)} + C</math> |
== See also == | == See also == | ||
Revision as of 15:54, January 1, 2009
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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.
