Difference between revisions of "Integral"

Revision as of 22:06, 15 November 2007

The integral is one of the two base concepts of calculus, along with the [[derivative.

There are two types of integrals:

Indefinite Integral

The indefinite integral, or antiderivative, is a partial inverse of the derivative. That is, if the derivative of a function $f(x)$ is written as $f'(x)$ , then the indefinite integral of $f'(x)$ is $f(x)+c$ , where $c$ is a real constant. This is because the integral of a constant is $0$ .

Notation

The integral of a function $f(x)$ is written as $\int f(x)\,dx$ , where the $dx$ means that the function is being integrated in relation to $x$ .
Often, to save space, the integral of $f(x)$ is written as $F(x)$ , the integral of $h(x)$ as $H(x)$ , etc.

Rules of Indefinite Integrals

$\int c\,dx=cx+C$ for a constant $c$ and another constant $C$ .
$\int f(x)+g(x)...+z(x)\,dx=\int f(x)\,dx+\int g(x)\,dx...+\int z(x)\,dx$
$\int x^n\,dx=\frac{1}{n+1}x^{n+1}+c$ , $n \ne -1$
$\int x^{-1}\,dx=\ln |x|+c$
$\int \sin x\,dx = -\cos x + c$
$\int \cos x\,dx = \sin x + c$
$\int\tan x\,dx = \ln |\cos x| + c$
$\int \sec x\,dx = \ln |\sec x + \tan x| + c$
$\int \csc \, dx =\ln |\csc x + \cot x| + c$
$\int \cot x\,dx = \ln |\sin x| + c$
$\int cx\, dx=c\int x\,dx$

Definite Integral

The definite integral is also the area under a curve between two points $a$ and $b$ . For example, the area under the curve $f(x)=\sin x$ between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ is $0$ , as are below the x-axis is taken as negative area.

Definition and Notation

The definite integral of a function between $a$ and $b$ is written as $\int^{b}_{a}f(x)\,dx$ .
$\int^{b}_{a}f(x)\,dx=F(b)-F(a)$ , where $F(x)$ is the antiderivative of $f(x)$ . This is also notated $\int f(x)\,dx \eval^{b}_{a}$ (Error compiling LaTeX. Unknown error_msg), read as "The integral of $f(x)$ evaluated at $a$ and $b$ ." Note that this means in definite integration, one need not add a constant, as the constants from the functions cancel out.

Rules of Definite Integrals

$\int^{b}_{a}f(x)\,dx=\int^{b}_{c}f(x)\,dx+\int^{c}_{a}$ for any $c$ .

Other uses

The word integral is the adjectival form of the noun "integer." Thus, $3$ is integral while $\pi$ is not.

@@ Line 1: / Line 1: @@
-The '''integral''' is a generalization of [[area]].  The integral of a [[function]] is defined as the area between it and the <math>x</math>-axis.  If the function lies below the <math>x</math>-axis, then the area is negative. It is also defined as the [[antiderivative]] of a function.
+The '''integral''' is one of the two base concepts of [[calculus]], along with the [[derivative.
-==Basic integrals==
-<math>\int x^n =\dfrac{x^{n+1}}{n+1}</math>
-<math>\int_{a}^{b} f'(x)= f(b)-f(a)</math>
-==Properties of integrals==
-<math>\int_{a}^b f = \int_a^c f + \int_c^b f</math>
+There are two types of integrals:
+==Indefinite Integral==
+The indefinite integral, or antiderivative, is a partial [[inverse]] of the [[derivative]]. That is, if the  derivative of a [[function ]]<math>f(x)</math> is written as <math>f'(x)</math>, then the indefinite integral of <math>f'(x)</math> is <math>f(x)+c</math>, where <math>c</math> is a [[real]] [[constant]]. This is because the integral of a constant is <math>0</math>.
+===Notation===
+*The integral of a function <math>f(x)</math> is written as <math>\int f(x)\,dx</math>, where the <math>dx</math> means that the function is being integrated in relation to <math>x</math>.
+*Often, to save space, the integral of <math>f(x)</math> is written as <math>F(x)</math>, the integral of <math>h(x)</math> as <math>H(x)</math>, etc.
+===Rules of Indefinite Integrals===
+*<math>\int c\,dx=cx+C</math> for a constant <math>c</math> and another constant <math>C</math>.
+*<math>\int f(x)+g(x)...+z(x)\,dx=\int f(x)\,dx+\int g(x)\,dx...+\int z(x)\,dx</math>
+*<math>\int x^n\,dx=\frac{1}{n+1}x^{n+1}+c</math>, <math>n \ne -1</math>
+*<math>\int x^{-1}\,dx=\ln |x|+c</math>
+*<math>\int \sin x\,dx = -\cos x + c</math>
+*<math>\int \cos x\,dx = \sin x + c</math>
+*<math>\int\tan x\,dx =  \ln |\cos x| + c</math>
+*<math>\int \sec x\,dx = \ln |\sec x + \tan x| + c</math>
+*<math>\int \csc \, dx =\ln |\csc x + \cot x| + c</math>
+*<math>\int \cot x\,dx = \ln |\sin x| + c</math>
+*<math>\int cx\, dx=c\int x\,dx</math>
+==Definite Integral==
+The definite integral is also the [[area]] under a [[curve]] between two [[points]] <math>a</math> and <math>b</math>. For example, the area under the curve <math>f(x)=\sin x</math> between <math>-\frac{\pi}{2}</math> and <math>\frac{\pi}{2}</math> is <math>0</math>, as are below the x-axis is taken as negative area.
+===Definition and Notation===
+*The definite integral of a function between <math>a</math> and <math>b</math> is written as <math>\int^{b}_{a}f(x)\,dx</math>.
+*<math>\int^{b}_{a}f(x)\,dx=F(b)-F(a)</math>, where <math>F(x)</math> is the antiderivative of <math>f(x)</math>. This is also notated <math>\int f(x)\,dx \eval^{b}_{a}</math>, read as "The integral of <math>f(x)</math> evaluated at <math>a</math> and <math>b</math>." Note that this means in definite integration, one need not add a constant, as the constants from the functions cancel out.
+===Rules of Definite Integrals===
+*<math>\int^{b}_{a}f(x)\,dx=\int^{b}_{c}f(x)\,dx+\int^{c}_{a}</math> for any <math>c</math>.
 ==Other uses==
@@ Line 21: / Line 36: @@
 *[[Chain Rule]]
-{{stub}}
 [[Category:Calculus]]
 [[Category:Definition]]

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