Math
Exercise 13
Choose the best answer.
Evaluate the following and choose either the correct answer or the option closest to the correct answer:
1. 
(a) |
x3 + C |
(b) |
|
(c) |
2x + C |
(d) |
 |
(b) 
Follow this method:
1) Ad 1 number to the power value and write down the x
2) Now adjust for the number in front of the x, taking the derivative
2. 
(a) |
-cos x + C |
(b) |
cos x + C |
(c) |
 |
(d) |
tan x + C |
(a) -cos x + C
See Rule 3, check taking the derivative
3. 
(a) |
 |
(b) |
|
(c) |
 |
(d) |
 |
(d)
Remember: 1/√x = 1x- ½ and x- ½ = x- 1 ½ -½
4. 
(a) |
–3 sin 3x + C |
(b) |
|
(c) |
sin 3x + C |
(d) |
-sin 3x + C |
(b) 
Substitute u for 3x, so you get
∫ cos 3x dx = cos u dx = sin u = 1/3 sin 3x + C
5. 
(a) |
 |
(b) |
|
(c) |
 |
(d) |
 |
(c)
See Rule 2.
6. 
(a) |
e2x + C |
(b) |
2e2x + C |
(c) |
 |
(d) |
 |
(c)
∫ e2x(½) = e2x(½) + C (use Rule 6)
7. 
(a) |
 |
(b) |
|
(c) |
 |
(d) |
 |
(b) 
Again, substitute u = 3x -4
∫ (3x-4) – 1 = ∫ (u) – 1
= ln |u| + C = ln (3x -4) 1/3 + C
8. Pollution from a factory is entering a lake. The rate of concentration of the pollutant at time t is given by 
where t is the number of years since the factory started introducing pollutants into the lake. Ecologists estimate that the lake can accept a total level of pollution of 4850 units, before all of the fish life in the lake ends. How long (in years) can the factory operate before all the fish are killed?
(a) |
1 |
(b) |
2 |
(c) |
4 |
(d) |
8 |
(c) 4
Rate of concentration at time t = P’(t) = 140 t 5/2
P(max) = 4850 = P(t)
∫140 t 5/2 = 40 t 7/2
4850 = 40 t 7/2, so t = 4
9. Solve the following initial value problem for y. 
; y(1)=5
(a) |
 |
(b) |
|
(c) |
 |
(d) |
 |
(c)
∫ x- 2 = - x- 1
For y(1) = 5, this means that 5 = - (1) - 1 + C ⇒ C = 6
10. Suppose that the skid marks of a stopping car are 160 feet long. The brakes created a constant deceleration of 20 feet/second2 ( 
) during the 4 seconds it took to stop. How fast was the car travelling (in feet per second) when the brakes were first applied?
(a) |
60 |
(b) |
80 |
(c) |
90 |
(d) |
100 |
(b) 80
Logical thinking: 4 x 20 feet = 80 feet
Evaluate the following:
11. 
(a) |
1 |
(b) |
5 |
(c) |
13 |
(d) |
20 |
(c) 13
Let u be (3x + 4) and integrate to 1/3 du = dx.
Now ∫ u 1/2 du can be integrated using rule number 1.
Substitute in the end u back to (3x +4).
You will get: 2/9 ( 3(7) + 4 ) 1.5 - 2/9 ( 3(4) + 4 ) 1.5 = 13
12. 
(a) |
0.25 |
(b) |
0.3 |
(c) |
0.5 | (d) |
0.8 |
(a) 0.25
∫ ½ sin 2x dx (check trigonometry for this rule)
= [ - ¼ cos 2x] = 0 + ¼(1)
13. 
(a) |
0.2 |
(b) |
0.5 |
(c) |
2 |
(d) |
5 |
(b) 0.5
½ tan 2 ( Π/8) – ½ tan 2 (0) = 0.5
14. 
(a) |
0.8 |
(b) |
1.5 |
(c) |
3.1 |
(d) |
7.5 |
(a) 0.8
Simplify the equation: ∫ x3 (1 + 2x + x2) dx = [ ¼ x4 + 2/5 x5 + 1/6 x6 ]
15. 
(a) |
1 |
(b) |
5 |
(c) |
9 |
(d) |
20 |
(c) 9
Again, simplify the equation to ∫ (x2 + x-2 ) dx = [1/3 x3 - x-1]
16. 
(a) |
1 |
(b) |
3 |
(c) |
10 |
(d) |
15 |
(b) 3
Notice that the root can be written as (….)1/3
Also notice that this means that the first part is the derivative of the second;
1/3 ( 6 + 3t2) = 3 + t2.
So this equation can be solved by substitution.
17. 
(a) |
0.25 |
(b) |
0.5 |
(c) |
0.7 |
(d) |
0.8 |
(a) 0.25
Use: cos dx = dsin x: ∫ cos2x sin x d sin x =
Use Pythagorean Identities: ∫ (1-sin2x) sin x d sin x
Now, substitute sin x = y, thus x = sin-1y
For x = 0, this means y = 0
For x = π/2, this means y = 1
Now you can state: from y = 0 to y = 1 :
∫ (1-y2) y d y = [ ½ y2- ¼ y4] = ½ - ¼
