Ch. 4‐ Antiderivatives Indefinite Integrals · 2014-02-07 · 4.1Antiderivatives and Indefinite Integrals.notebook 14 February 07, 2014 Initial Conditions and Particular Solutions

4.1Antiderivatives and Indefinite Integrals.notebook

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Ch. 4‐Antiderivatives

& Indefinite Integrals


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Theorem:If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is

G(x) = F(x) + Cwhere C is a constant.


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G(x) = F(x) + C

• C is called the constant of integration

• G is the general antiderivative of f

• G(x) = F(x) + C is the general solution of the differential equation G '(x) = F '(x) = f(x)

• A differential equation in x and y is an equation that involves x, y, and derivatives of y. (y' = 3x)


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Example:Find the general solution of the differential equation y' = 2.In other words, find the original equation that gives you this derivative.


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Example: Find the antiderivative of y = 2x.


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dydx = f (x)

When solving a differential equation of the form

it is convenient to write it in the equivalent differential form

dy = f(x) dx.

The operation of finding all solutions of this equation is called antidifferentiation (or indefinite integration) and is denoted by an integral sign ∫.


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variable of integration

integrand

constant of integration

antiderivative of f(x)

y =

Notations of Antiderivatives

The expression is read as the anderivave of f with respect to x. The differenal dx serves to idenfy x as the variable of integraon. The term indefinite integral is a synonym for anderivave.

The inverse nature of integration and differentiation can be verified by substituting F'(x) for f(x) in the indefinite integration definition to obtain

Moreover, if ∫f(x)dx = F(x) + C, then

These two equations allow you to obtain integration formulas directly from differentiation formulas, as shown in the following summary.


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Basic Integration Rules

Log Function:

Natural Exponential Function:

Exponential Function:


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Formulas to know!!! MEMORIZEFunction Particular Antiderivative


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Examples:


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More Examples:


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More Examples:

*Rewrite the function when necessary.


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Initial Conditions and Particular SolutionsThe equation y = ∫f(x)dx has many solutions (each differing from the others by a constant).

This means that the graphs of any two antiderivatives of f are vertical translations of each other.

In many applications of integration, you are given enough

information to determine a particular solution. To do this,

you need only know the value of y = F(x) for one value of x.

This information is called an initial condition.

F(x) = x3 – x + C General solution

F(2) = 4 Initial condition

Using the initial condition that F(2)=4, find the equation that passes through this point. This equation is the particular solution.


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Example:

1. Write a function that could have the derivative:

Is this the only possibility?

2. Assume that (1, 1) is a point on the graph of the function.

How is this added information helpful?


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Particle Motion Example:A particle moves in a straight line and has acceleration given by

Its initial velocity is Its initial displacement isFind its position function, s(t).

Ch. 4‐ Antiderivatives Indefinite Integrals · 2014-02-07 · 4.1Antiderivatives and Indefinite Integrals.notebook 14 February 07, 2014 Initial Conditions and Particular Solutions

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