Antiderivatives Indefinite Integrals

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

Basic Integration Rules Rules:

Applying the Basic Integration Rules  Evaluate the indefinite integral

Rewriting Before Integrating

Integrating Polynomial Functions

Integrating Polynomials

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

Rewriting Before Integrating

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.

Finding a Particular Solution

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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