Difference between revisions of "User:Temperal/The Problem Solver's Resource10"

Revision as of 17:36, 13 October 2007

The Problem Solver's Resource

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 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

$\int c\,dx=0$ for a constant $c$ .
$\int f(x)+g(x)...+z(x)\,dx=\int f(x)\,dx+\int g(x)\,dx...+\int z(x)\,dx$
$\int x^n\,dx=\frac{1}{n+1}x^{n+1}+c$ , $n \ne -1$
$\int x^{-1}\,dx=\ln |x|+c$
$\int \sin x\,dx = -\cos x + c$
$\int \cos x\,dx = \sin x + c$
$\int\tan x\,dx = \ln |\cos x| + c$
$\int \sec x\,dx = \ln |\sec x + \tan x| + c$
$\int \csc \, dx =\ln |\csc x + \cot x| + c$
$\int \cot x\,dx = \ln |\sin x| + c$

Back to page 9 | Continue to page 11

@@ Line 17: / Line 17: @@
 *<math>\int c\,dx=0</math> for a 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>
@@ Line 23: / Line 25: @@
 *<math>\int \csc \, dx =\ln |\csc x + \cot x| + c</math>
 *<math>\int \cot x\,dx = \ln |\sin x| + c</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>
 [[User:Temperal/The Problem Solver's Resource9|Back to page 9]] | [[User:Temperal/The Problem Solver's Resource11|Continue to page 11]]
 |}<br /><br />

Page

Toolbox