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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

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

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 | style="background:lime; border:1px solid black;height:200px;padding:10px;" | {{User:Temperal/testtemplate|page 10}}
 ==<span style="font-size:20px; color: blue;">Integrals</span>==
-<!-- will fill in later! -->
+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 <math>f(x)</math> is written as <math>f'(x)</math>, then the indefinite integral of <math>f'(x)</math> is <math>f(x)+c</math>, where <math>c</math> is a real constant. This is because the integral of a constant is <math>0</math>.
+====Notation====
+*The integral of a function <math>f(x)</math> is written as <math>\int f(x)\,dx</math>, where the <math>dx</math> means that the function is being integrated in relation to <math>x</math>.
+*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 f(x)+g(x)...+z(x)\,dx=\int f(x)\,dx+\int g(x)\,dx...+\int z(x)\,dx</math>
+*<math>\int \sin x\,dx = -\cos x + c</math>
+*<math>\int \cos x\,dx = \sin x + c</math>
+*<math>\int\tan x\,dx =  \ln |\cos x| + c</math>
+*<math>\int \sec x\,dx = \ln |\sec x + \tan x| + c</math>
+*<math>\int \csc \, dx =\ln |\csc x + \cot x| + c</math>
+*<math>\int \cot x\,dx = \ln |\sin 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]]
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