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?
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
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.
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.
Differential Equations Find the general solution of the differential equation y’ = x 2 Find the general solution of the differential equation
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)
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.
Basic Integration Rules Differentiation and antidifferentiation are inverse functions (one “undoes” the other) The basic integration rules come from the basic derivative rules
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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