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Line 1: − 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 integration. However, the [[fundamental theorem of calculus]] relates a [[definite integral]] to an indefinite integral by taking its value at the boundary points. − + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Indefinite integral (view source)
Revision as of 21:49, December 30, 2008
, 21:49, December 30, 2008added a list of simple integrals
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.
== See Also ==
== 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, <math>C</math>, is always added.
== A list of simple antiderivatives ==
The identity antiderivative:
:<math>\int dx = x + C</math>
=== Polynomial and simple rational ===
:<math>\int xdx = {1 \over 2} x^2 + C</math>
:<math>\int x^2dx = {1 \over 3} x^3 + C</math>
:<math>\int {1 \over {x^3}}dx = \int x^{-3}dx = {1 \over -3}x^{-2} + C = {-1 \over {3x^{-2}}} + C</math>
:<math>\int x^{3.873}dx = {1 \over 4.873} x^{4.873} + C</math>
The general rule for polynomial expressions is:
:<math>\int x^ndx = {1 \over {n+1}} x ^ {n + 1} + C</math>
Note: <math>n \ne -1</math>. See below for when <math>n=-1</math>
=== Rational ===
{{main|Partial fractions in integration}}
Rational antiderivatives are much more difficult and follow different rules.
: <math>\int {1 \over x}dx = \int x^{-1}dx = \ln(x) + C</math>
: <math>\int {1 \over {x+1}}dx = \ln(x+1) + C</math>
: <math>\int {1 \over {x+a}}dx = \ln(x+a) + C</math>
=== Trigonometric ===
:<math>\int \sin(x)dx = -\cos(x) + C</math>
:<math>\int \cos(x)dx = \sin(x) + C</math>
:<math>\int \tan(x)dx = \ln|\sec(x)| + C</math>
:<math>\int \sin(x)dx = -\cos(x) + C</math>
=== Exponential ===
:<math>\int e^x = e^x + C</math>
:<math>\int a^x = \ln(a)a^x + C</math>
== See also ==
*[[Integral]]
*[[Integral]]