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§4.2 Integration Using Logarithmic and Exponential Functions.

§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. ∫ x 3 dx =.

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§4.2 Integration Using Logarithmic and Exponential Functions.

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  1. §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.

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

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

  4. Examples – INTEGRATING EXPONENTIAL FUNCTIONS

  5. Application 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.

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

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

  8. 1. ∫ 1/x dx = ln |x| + C, x ≠ 0. ∫ x - 1 dx = Examples 2. ∫ 4/x dx = 4 ∫ 1/x dx = 4 ln |x| + C

  9. 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

  10. S (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. But S (1) = 0 & ln |1|= 0 so Part 2 And C = 0 so S (t) = 60 ln | t |

  11. 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

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

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