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Antiderivatives Indefinite Integrals

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Definition A function F is an antiderivative of f on an interval I if F’(x) = f(x) for all x in I. Example: F(x) = x 3 when f(x) = 3x 2 In other words, we are going backwards – given the derivative, what was the original function?

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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 iff G(x) = F(x) + C, for all x in I where C is a constant

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Representation of Antiderivatives The constant C is called the constant of integration. The family of functions represented by G is the general antiderivative of f and the answer is the general solution of the differential equation.

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Differential Equation A differential equation in x and y is an equation that involves x, y, and derivatives of y. Find the general solution of the differential equation y’ = 2.

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Differential Equations Find the general solution of the differential equation y’ = x 2 Find the general solution of the differential equation

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Notation for Antiderivatives When solving a differential equation of the form The operation of finding all solutions of this equation is called antidifferentiation (or indefinite integration)

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Notation for Antiderivatives The general solution is denoted by Where f(x) is the integrand, dx is the variable of integration and C is the constant of integration. This is an antiderivative of f with respect to x.

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Basic Integration Rules Differentiation and antidifferentiation are inverse functions (one “undoes” the other) The basic integration rules come from the basic derivative rules

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Basic Integration Rules Rules:

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Applying the Basic Integration Rules Evaluate the indefinite integral

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Rewriting Before Integrating

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Integrating Polynomial Functions

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Integrating Polynomials

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Rewriting Before Integrating We have no division rules at this point When dividing by a monomial, put the monomial under each of the monomials in the numerator

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Rewriting Before Integrating

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Initial Conditions and Particular Solutions A particular solution allows you to determine the C in a differential equation. In order to do this, initial conditions must be given.

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Finding a Particular Solution

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Solving a Vertical Motion Problem A ball is thrown upward with an initial velocity of 64 feet per second from an initial height of 80 feet a. Find the position function giving the height s as a function of the time t. b. When does the ball hit the ground?

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On-line Help Tutorial Tutorial Video Help Video Help Video Help

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