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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Subjects
Calculus II
Fundamental Theorem of Calculus
With both limits variable:

3.3. Fundamental Theorem of Calculus

With both limits variable:

In 3.3. Fundamental Theorem of Calculus

Formula

\frac{d}{d x} \int_{u (x)}^{v (x)} f (t) d t = f (v (x)) v^{'} (x) - f (u (x)) u^{'} (x) .

Variables

$x$ Variable independiente (variable de integración)
$u$ Variable auxiliar de sustitución
$t$ Parámetro o variable auxiliar
$f$ Función integrando
$d x$ Diferencial respecto de $x$
$v$ Función o diferencial en integración por partes ( $d v$ )

Related formulas

Related
More generally, by the chain rule:
$\frac{d}{d x} \int_{a}^{g (x)} f (t) d t = f (g (x)) g^{'} (x) .$
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Related
Formula
$\int_{a}^{b} f (x) d x = F (b) - F (a) .$
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Related
Formula
$G^{'} (x) = f (x) .$
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Related
Formula
$G (x) = \int_{a}^{x} f (t) d t,$
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