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

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

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

4. ∫ 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

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

6. 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

7. 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

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

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

10. 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

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

12. 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. 13. BUSINESS: Cost – The weekly marginal cost of producing shoes is given by C’ (x) = /(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. 14. BUSINESS: Price-demand – The marginal price p’ (x) at x boxes of a certain cereal per week is given by p’ (x) = e -.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 = x then du = and the integral becomes
And integration yields
Step 2: Find p (x). At a price of $4.35, 50 boxes are sold, so

15. 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 = t then du = 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)

16. 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

17. 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.