User:Temperal/The Problem Solver's Resource10

Introduction \| Other Tips and Tricks \| Methods of Proof \| You are currently viewing page 10.

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

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 $\int_a^b F'(x)dx = F(b) - F(a)$ .