Mathematical formulas

Calculus II

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Sections

1Notation, domains and conventions
2Indefinite integral and properties
3Definite integral and Fundamental Theorem of Calculus
4Substitution method
5Integration by parts
6Trigonometric integrals
7Trigonometric substitution and quadratic radicals
8Rational functions and partial fractions
9Improper integrals
10Applications of the integral
11Numerical integration
12Parametric curves
13Polar coordinates
14Sequences and series
15Power series and Taylor series
16Elementary differential equations
17Guide to choosing a method
AAppendix A: extensive table of antiderivatives
BAppendix B: useful equivalences

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Calculus II
Term-by-term differentiation and integration
Formula

15.2. Term-by-term differentiation and integration

Formula

In 15.2. Term-by-term differentiation and integration

Formula

\int\sum_{n=0}^{\infty}c_n(x-a)^n\,dx =C+\sum_{n=0}^{\infty}\frac{c_n}{n+1}(x-a)^{n+1}.

Tags

maclaurin
series
series powers Taylor differentiation and integration term A term
Taylor

Variables

$C$ Constante real de integración
$x$ Variable independiente (variable de integración)
$a$ Constante real (a menudo $a>0$ )
$n$ Exponente o índice entero
$dx$ Diferencial respecto de $x$

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Related formulas

Related
Within $|x-a|<R$ :
$\frac{d}{dx}\sum_{n=0}^{\infty}c_n(x-a)^n =\sum_{n=1}^{\infty}nc_n(x-a)^{n-1}$
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