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
Fundamental Theorem of Calculus
Formula

3.3. Fundamental Theorem of Calculus

Formula

In 3.3. Fundamental Theorem of Calculus

Formula

\int_a^b f(x)\,dx=F(b)-F(a).

Tags

antiderivative
integral definida
integral definite FTC theorem fundamental of calculus
property
teorema fundamental

Variables

$x$ Variable independiente (variable de integración)
$a$ Constante real (a menudo $a>0$ )
$b$ Real constant
$f$ Función integrando
$F$ Antiderivada de $f$
$dx$ Diferencial respecto de $x$

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The indefinite integral represents the family of antiderivatives:
$\int f(x)\,dx=F(x)+C.$
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With both limits variable:
$\frac{d}{dx}\int_{u(x)}^{v(x)}f(t)\,dt =f(v(x))v'(x)-f(u(x))u'(x).$
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More generally, by the chain rule:
$\frac{d}{dx}\int_a^{g(x)}f(t)\,dt=f(g(x))g'(x).$
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$G'(x)=f(x).$
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$G(x)=\int_a^x f(t)\,dt,$
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