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The Problem Solver's Resource

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

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@@ Line 15: / Line 15: @@
 *Often, to save space, the integral of <math>f(x)</math> is written as <math>F(x)</math>, the integral of <math>h(x)</math> as <math>H(x)</math>, etc.
 ====Rules of Indefinite Integrals====
-*<math>\int c\,dx=0</math> for a constant <math>c</math>.
+*<math>\int c\,dx=cx+C</math> for a constant <math>c</math> and another 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>

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