§4.2 Integration Using Logarithmic and Exponential Functions. The student will learn about: the integral of exponential functions, the integral of logarithmic functions, and some applications for each of these functions.

First Some Practice a. ∫ x3 dx = b. ∫ (x 4 + x + x ½ + 1 + x – ½ + x – 2) dx =

Integral of Exponential Functions. 1. ∫ e x dx = e x + C. 2. Notice, this is just the original function divided by a!

Examples – INTEGRATING EXPONENTIAL FUNCTIONS

Application f (t) = f (t) = 375 e 0.04 t + C World consumption of copper is running at a rate of 15 e 0.04 t million metric tons per year where t = 0 corresponds to 2000. Find a formula for the total amount of copper that will be used within t years of 2000. We are given the growth rate and to find the total we integrate that rate. a = 0.04 Part 1 f (t) = f (t) = 375 e 0.04 t + C Note that there is no part 2 or part 3 in this problem.

Integral of Logarithmic Functions. 1. ∫ 1/x dx = ln |x| + C, x ≠ 0. ∫ x - 1 dx = Be careful! Don’t confuse this special case with the power rule. Power rule. logarithmic rule.

The Integral The integral can be written in three different ways, all of which have the same answer.

Examples 1. ∫ 1/x dx = ln |x| + C, x ≠ 0. ∫ x - 1 dx = 2. ∫ 4/x dx =

Application The College Bookstore is running a sale on the most popular mathematics text. The sales rate [books sold per day] on day t of the sale is predicted to be 60/t ( for t ≥ 1) where t = 1 represents the beginning of the sale, at which time none of the inventory of 350 books has been sold. Find a formula for the number of books sold up to day t. We are given the sales rate and to find the total sales we integrate that rate. S (t) = Part 1

Application - Continued The College Bookstore is running a sale on the most popular mathematics text. The sales rate [books sold per day] on day t of the sale is predicted to be 60/t ( for t ≥ 1) where t = 1 represents the beginning of the sale, at which time none of the inventory of 350 books has been sold. S (t) = Find C. We need to first find C using the initial point t = 1 S = 0. S (1) = But S (1) = 0 & ln |1|= 0 so Part 2 And C = 0 so S (t) = 60 ln | t |

Application - Continued The College Bookstore is running a sale on the most popular mathematics text. The sales rate [books sold per day] on day t of the sale is predicted to be 60/t ( for t ≥ 1) where t = 1 represents the beginning of the sale, at which time none of the inventory of 350 books has been sold. S (t) = Will the store have sold its inventory of 350 books by day t = 30? Let t = 30. S (30) = 60 ln | 30 | Part 3 = (60) (3.40) = 204 NO. The bookstore will still have 350 – 204 = 146 books in stock. 11

ASSIGNMENT §4.2 on my website. 12, 13, 14.