ANTIDERIVATIVES AND INDEFINITE INTEGRATION The opposite of a derivative is called an antiderivative or integral. Definition: A function F is an antiderivative of f on an interval I if for all x in I. EX #1: Antiderivatives differ only by a constant, C: C is called the constant of integration Family of all antiderivatives of f(x) = 2x and the general solution of the differential equation A differential equation in x and y is an equation that involves x, y, and derivatives of y. For example: and 1 EX #2: Solving a Differential Equation EX #3: Notation for antiderivatives: The operation of finding all solutions of this equation is called antidifferentiation or indefinite integration denoted by sign. differential form General Solution is denoted by: Variable of Constant of Integration Integration Integrand read as the antiderivative of f with respect to x. So, the differential dx serves to identify x as the variable of integration. The term indefinite integral is a synonym for antiderivative. 2 BASIC INTEGRATION RULES Integration is the “inverse” of differentiation. Differentiation is the “inverse” of integration. Differentiation Formula Integration Formula POWER RULE: 3 EX #4: Applying Basic Rules EX #5: Rewriting Before Integrating Original Integral Rewrite Integrate Simplify 4 EX # 6: Polynomial Functions A. B. C. EX #7: Integrate By Rewriting 5 EX #8: Solve differential equations subject to given conditions. Given: and 6 EX #9: 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? 7 EX. #10: An evergreen nursery usually sells a certain shrub after 6 years of growth and shaping. The growth rate during those 6 years is approximated by dh/dt = 1.5t + 5, where t is the time in years, and h is the height in centimeters. The seedlings are 12 centimeters tall when planted (t = 0). A. Find the height after t years. [Hint: the derivative is a rate of change of a function and the integral is the initial function.] B. How tall are the shrubs when they are sold? 8.