Mathematical formulas

Calculus II

Subjects Home Search Guide

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
Formula

3.3. Fundamental Theorem of Calculus

Formula

In 3.3. Fundamental Theorem of Calculus

Formula

\int_{a}^{b} f (x) d x = F (b) - F (a) .

Variables

$x$ Variable independiente (variable de integración)
$a$ Constant real (a menudo $a > 0$ )
$b$ Constant real
$f$ Función integrando
$F$ Antiderivada de $f$
$d x$ Diferencial respecto de $x$

Related formulas

Related
The indefinite integral represents the family of antiderivatives:
$\int f (x) d x = F (x) + C .$
Open formula Open in section
Related
With both limits variable:
$\frac{d}{d x} \int_{u (x)}^{v (x)} f (t) d t = f (v (x)) v^{'} (x) - f (u (x)) u^{'} (x) .$
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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
$G^{'} (x) = f (x) .$
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Related
Formula
$G (x) = \int_{a}^{x} f (t) d t,$
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