Integral

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

Beginner Level

In introductory, high-school level texts, the integral is often presented in two parts, the indefinite integral and definite integral. Although this approach lacks mathematical formality, it has the advantage of being easy to grasp and convenient to use in most of its applications, especially in Physics.

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

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 area 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 $[\int f(x)\,dx]^{b}_{a}$ , 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

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

Also, $\text{Area} = \iint f(x, y)\,dA$ , $\text{Vol} = \iiint f(x, y, z)\,dV$ , $\text{4V} = \iiiint f(x, y, z, w)\,dV$ .

Formal Use

The notion of an integral is one of the key ideas in severel areas of higher mathematics including analysis and topology. The integral can be defined in several ways which can be applied to several different settings. However, the most common definition, and the one which most closely resembles the the 'definite integral' is the Riemann Integral

Riemann Integral

Let $f:[a,b]\rightarrow\mathbb{R}$

Let $L\in\mathbb{R}$

We say that $f$ is Riemann Integrable on $[a,b]$ if and only if

$\forall \epsilon>0\:\exists\delta>0$ such that if $\mathcal{\dot{P}}$ is a tagged partition on $[a,b]$ with $\|\mathcal{\dot{P}}\|<\delta$ $\implies$ $|S(f,\mathcal{\dot{P}})-L|<\epsilon$ , where $S(f,\mathcal{\dot{P}})$ is the Riemann sum of $f$ with respect to $\mathcal{\dot{P}}$

$L$ is said to be the integral of $f$ on $[a,b]$ and is written as $L=\int_a^b f(x)dx=\int_a^b f$

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Another integral commonly used in introductory texts is the Darboux Integral (which is often called the Riemann Integral)

Darboux Integral

Let $f:[a,b]\rightarrow\mathbb{R}$

We say that $f$ is Darboux Integrable on $[a,b]$ if and only if $\inf_{\mathcal{P}}U(f,\mathcal{P})=\sup_{\mathcal{P}}L(f,\mathcal{P})$ , where $L(f,\mathcal{P})$ and $U(f,\mathcal{P})$ are respectively the lower sum and upper sum of $f$ with respect to partition $\mathcal{P}$

The notation used for the Darboux integral is the same as that for the Riemann integral.

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Other Definitions

Other important definitions of integration include the Riemann-Stieltjes integral, Lebesgue integral, Henstock-Kurzweil integral, etc.

Disambiguation

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

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