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4.1 Antiderivatives and Indefinite Integrals Defn. A function F(x) is an antiderivative of f(x) on an interval I if F '(x)=f(x) for all x in I. ex. Find an antiderivative of f(x)=x 2 ex. Find an antiderivative of f(x)=sinx ex. Find an antiderivative of f(x)=

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Thm. If F is an antiderivative of f on an interval I, then G is an antiderivative of f iff G is of the form G(x)=F(x)+c, where c is a constant. G is called a General Antiderivative. c is called the Constant of Integration. Defn. A Differential Equation in x and y is an equation that involves derivatives of y. These all mean the same thing: Find the general solution of a differential equation. Find an antiderivative of a function. Evaluate an indefinite integral. So… how many antiderivatives of f are there? An infinite number

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ex. Find a general solution for the differential equation y'=3x. ex. Find the particular solution for the differential equation y'=3x if x=1 when y=2/3.

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Notation = f '(x) dy= f '(x) dx This is an antiderivative symbol, also called an integral symbol. It's an elongated S. } integrand This shows the variable of integration. } The antiderivative The constant of integration. Sometimes the integrand needs to be rewritten to be more useful.

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Defn. The operation of finding antiderivatives is called indefinite integration. Basic Integration Rules (p.250) So, integration and differentiation are inverse operations (just like + and -, x and ÷, or squaring and square rooting). For some constant k,

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If dy/dx = 2x-1 what is the general solution of this differential equation? What is the particular solution of this differential equation for x=1 and y=1? Suppose that the acceleration of a particle is a(t)= Find its position function x(t) if v(4)=2 and x(0)=0. HW: p256#3-42x3

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