Changes
added a list of simple integrals
An '''indefinite integral''', or antiderivative, is an [[integral]] without upper and lower limits.
== 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 Also 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]]
