Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

User:Temperal/The Problem Solver's Resource10

< User:Temperal
Revision as of 17:11, 17 October 2007 by Temperal (talk | contribs) (→‎Rules of Indefinite Integrals: correct)



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

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

Back to page 9 | Continue to page 11



Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=User:Temperal/The_Problem_Solver%27s_Resource10&oldid=18273"