operaciones de potencias

8/11/2019 Operaciones de Potencias

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Operaciones de Potencias


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Operaciones de las Derivadas

En las siguientes frmulas de derivadas:

y representan funciones

es una constante (un nmero)

Operaciones

Frmulas


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Integracin de Funciones Logaritmo

Reglas para la integracin de funciones logaritmo

Se usan las barras de valor absoluto ya que el dominio de una funcinlogaritmo son los nmeros reales positivos.

En particular,

= 2 ln (x) + c

= ln (x2) + c

En este ejemplo, no es necesario las barras de valor absoluto ya que x2nopuede ser negativo.

Ejemplos para discusin: Halla:

Ejercicio de prctica: Halla:


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Ejercicios: Halla la integral:

Respuestas:

1) ln x + 1 + C2) ln x - 5 + C3) - ln 3 2x + C


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Ejemplos:

Propiedad #1: Todo numero elevado a 1 es 0.

Propiedad #2: Todo numero elevado por si mismo es 1.
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Propiedad #3:

Propiedad #4:
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Practice problems for these concepts can be found at:Integration Practice Problems for AP

Calculus

Antiderivatives and Integration Formulas

Definition: A function Fis an antiderivative of another function fif F'(x) = f(x) for all xin

some open interval. Any two antiderivatives of fdiffer by an additive constant C. We

denote the set of antiderivatives of fby f(x)dx, called the indefinite integral of f.

Integration Rules:

1. f(x)dx= F(x) + C F'(x) = f(x)

2. af(x)dx= a f(x)dx

3. f(x)dx= f(x)dx

4. [f(x) g(x)] dx= f(x)dx g(x)dx
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20. ln xdx= xln |x|x+ C

21.

22.

23.

24.

Note: After evaluating an integral, always check the result by taking the derivative of

the answer (i.e., taking the derivative of the antiderivative).

Evaluating Integrals


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Example 1

Evaluate (x56x2+ x1) dx.

Applying the formula .

Example 2

Evaluate .


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Rewrite

Example 3

If = 3x2+ 2, and the point (0, 1) lies on the graph of y, find y.

Since = 3x2+ 2, then yis an antiderivative of . Thus, y= (3x2+ 2) dx= x3+

2x+ C. The point (0, 1) is on the graph of y.

Thus, y= x3 + 2x+ Cbecomes 1 = 03+ 2(0) + Cor C= 1. Therefore, y= x3+ 2x

1.

Example 4

Example 5

Example 6


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Example 7

Example 8

Evaluate (4 cos xcot x) dx.

(4 cos xcot x) dx= 4 sin xln |sin x| + C.

Example 9

Example 10

Example 11

Example 12


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Example 13

operaciones de potencias

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