Section 4.3 Let u = x 2 +1. Then du = 2x dx and the integral becomes And reversing the substitution yields:

∫ e u du = e u + C Let u = x 5 . Then du = 5x4 dx and the integral becomes And reversing the substitution yields:

Let u = x 4 - 16. Then du = 4x3 dx and we need to make an adjustment for the 4. And then the integral becomes

∫ e u du = e u + C Let u = 3x. Then du = 3 dx and we need to make an adjustment for the 3. And then the integral becomes

Let u = 1 + 5x. Then du = 5 dx and we need to make an adjustment for the 5. And then the integral becomes

Let u = x 4 + 16. Then du = 4 x 3 dx and we need to make an adjustment for the 4. And then the integral becomes

Let u = 2x 2 + 4x. Then du = 4x +4 dx = 4(x + 1) dx and we need to make an adjustment for the 4. And then the integral becomes

Let u = 3x 4 + 4x 3 . Then du = 12x 3 + 12x 2 dx = 12(x 3 + x 2) dx and we need to make an adjustment for the 12. And then the integral becomes

Let u = 1 – x 2. Then du = - 2x dx and we need to make an adjustment for the - 2. And then the integral becomes

Let u = e 2x + 1. Then du = 2e 2x dx and we need to make an adjustment for the 2. And then the integral becomes

Let u = ln x. Then du = 1/x dx. And then the integral becomes

This one is a little different. We need to do a little algebra first This one is a little different. We need to do a little algebra first. Distribute the x 2.

13. BUSINESS: Cost – The weekly marginal cost of producing shoes is given by C’ (x) = 12 + 500/(x + 1) where C (x) is the cost in dollars. If the fixed cost are $2,000 per week, find the cost function. Step 1: Integrate C’ to find C. Let u = x + 1 then du = dx and the integral becomes Step 2: Find C. The fixed cost are 2000 or C(x) = 2000 when x = 0.

14. BUSINESS: Price-demand – The marginal price p’ (x) at x boxes of a certain cereal per week is given by p’ (x) = - 0.015e -.01x . Find the price equation if the weekly demand is 50 boxes when the price is $4.35. Step 1: Integrate p’ to find p. Let u = - 0.01x then du = - 0.01 and the integral becomes And integration yields Step 2: Find p (x). At a price of $4.35, 50 boxes are sold, so

15. MEDICINE – The rate of healing for a skin wound (in square centimeters per day) is approximated by A’ (t) = - 0.9e -0.1t . If the initial wound has an area of 8 square centimeters, what will its area be after t days? After 5 days? Step 1: Integrate A’ to find A. Let u = - 0.1 t then du = - 0.1 dt and the integral becomes Step 2: Find C. The size of an initial wound is 8 when x = 0. Step 3. Find A (5)

POLLUTION - A contaminated lake is treated with a bactericide POLLUTION - A contaminated lake is treated with a bactericide. The rate of increase in harmful bacteria t days after the treatment is given by Where N (t) is the number of bacteria per milliliter of water. Find the minimum value of dN/dt. If the initial count was 5,000 bacteria per milliliter, find N (t). Find the bacteria count after 10 days. Use your calculator to minimize the given function. The minimum value is – 1000. b. Integrate dN/dt ’ to find N. Let u = 1 + t 2 then du = 2t dt and the integral becomes CONTINUED

Find C. The initial count was 5000 when t = 0. c. Find N (10). 10 days after treatment the bacteria count will be 385 bacteria per milliliter of water.