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User:Temperal/The Problem Solver's Resource10

< User:Temperal
Revision as of 16:05, 13 October 2007 by Temperal (talk | contribs) (→‎Rules of Indefinite Integrals)



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

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