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4.1Antiderivatives and Indefinite Integrals.notebook 1 February 07, 2014 Ch. 4‐ Antiderivatives & Indefinite Integrals
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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).
