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4.1 ANTIDERIVATIVES & INDEFINITE INTEGRATION

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Definition of Antiderivative A function is an antiderivative of f on an interval I if F’(x) = f(x) for all x in I Note: F is called an antiderivative of f, rather than the antiderivative of f because: F 1 (x)=x 3, F 2 (x)=x 3 – 5 and F 3 (x)=x 3 +97 are all antiderivatives of f(x)=3x 2

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Representation of Antiderivatives If F is an antiderivative of f on an interval I, then G is an antiderivative of f on the interval I if and only if G is of the form: G(x) = F(x) + c, for all x in I where c is a constant

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Example 1:Solving a differential Find the general solutions of the differential equation y’=2

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Notation for antiderivatives When solving a differential equation of the form: It is convenient to write it in the equivalent form: dy = f(x) dx The process of finding all solutions of this equation is called antidifferentiation (or indefinite integration)

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Notation for Antiderivatives Antidifferentiation is denoted by the integral sign: The general solution is denoted by: Integrand Variable of Integration Constant of Integration

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Basic Integration Rules Moreover. If, then

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Basic Integration Rules Look at rules on page 244

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Power Rule

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Example 1

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Example 2

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Example 3

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Example 4

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Example 5

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Example 6

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Example 7

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Example 8

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