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Integrals

This section will cover integrals and related topics, the Fundamental Theorem of Calculus, and some other advanced calculus topics.

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

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 $\int f(x)\,dx \eval^{b}_{a}$ , 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 $c$ .

Fundamental Theorem of Calculus

Let ${a}$ , ${b} \in \mathbb{R}$ , $a<b$ . Suppose $F:[a,b] \to \mathbb{R}$ is differentiable on the whole interval $[a,b]$ (using limits from the right and left for the derivatives at ${a}$ and ${b}$ , respectively), and suppose that $F'$ is Riemann integrable on $[a,b]$ . Then $\int_a^b F'(x)dx = F(b) - F(a)$ .