Basic Rules of Integration: Worksheet #1
Name:;_________________________________
Determine the Following:
1.
 e dx
2.
4.
x
5. sin xdx
7.
 csc x cot xdx
x
10.
2/3
 x 2/5 dx
5

1  25 x 2
 x dx
3.

2
6. sec xdx

9.
9
3
xdx

8. 5 cos xdx

x
1
dx
1
12.
 x dx

15.


18.
4e4 x
 1  e8 x dx
11. 12dx
dx

Determine the following. Beware of compensation.

14. sec(12 x) tan(12 x)dx
13. sin(8 x) dx
16.
e
18 x
17. 3cos(9 x) dx
dx
x
1. e  C
2.
1
10
x10  C
3.
3
4
x 4/3  C 4.
3
5
x 5/3  C 5.  cos x  C 6. tan x  C
2 x  1dx
7.  csc x  C 8.
5sin x  C 9. ln x  C 10. sin 1 (5 x)  C 11. 12x  C 12. ln x  C 13.  18 cos(8 x)  C 14.
1
3
(2 x  1)
3/2
 C 16.
1
18
18 x
e
 C 17. 13 sin(9 x)  C
1
18. tan (e )  C
4x
1
12
sec(12 x)  C 15.
Fundamental Theorem of Calculus Part 2
Name:_________________________________________
What is the difference between a definite integral and an indefinite integral?
Fundamental Theorem of Calculus Part 2 (Yes, we’re starting with Part 2)

Practice with Definite integrals
Worksheet #2- No Calculator
Name:_______________________________________
6
1.
3.
2
 x dx
2.
3
3
5
0
x
 e dx
4.
1
7.
  4t
du
  3x  2e  dx
x
 t 7/2  dt
4
1
3
t
10
4
3/2
2
2
3
5.
u
6.
2
dt
1
dx
.2 3x
Turn Over

2
8.  du
2
Worksheet # 2 (Continued)
 /2
t 1
dt
9. 
t
1
27
3 / 4
11.




10.
 /2
 /4
sin  d
 sec
12.
 /3


1/2
14.
csc x cot xdx
/6
11.
tdt
2
 1 4x
2
dx
0
5
3
1. 27/2 2. 35/3 3. e  e
60 3 
2
0
/4
13.
cos xdx
2
4. 2e  8
8
10. 2
3
2 12. 1
13. 2  2 3 3
14. π/4
5.
162 3 82

5
45
6. ¾
7. 1/3 ln50
2
8. 2 9.
U-Substitution Worksheet #3
Name:_______________________ Date:___
1.
x2
 (1  x3 )dx
8
2
2. cot  csc  d
3.

sin x
4. e cos xdx

4
tan  sec 2  d
5. sin(2 x  4)dx


6.
 x( x
2
 5)10 dx


10.
2
4
9. sec x tan xdx
11.
dx
 x ln x
3
1. 1/ 3ln 1  x  C
1
22
csc2 (ln x)
 x dx
8.
2
7. sin x cos xdx
e x dx
 (e x  1)3
x
 xe dx
2
12.
2.  19 cot 9   C 3.
4
5
tan 5/ 4   C 4. esin x  C 5.  12 cos(2 x  4)  C 6.
( x 2  5)11  C
7. 13 sin 3 x  C 8.  cot(ln x)  C
9.
1
5
tan 5 x  C 10.  12 (e x  1) 2  C 11. 2(ln x)1/2  C 12.
U-Substitution Definite Integrals: Worksheet #4
No Calculator.
x2
1 (2 x2  8x)2 dx
Name:____________________ Period:___
1
2
1.
2.
 /2
1

2
3.  cos( )d
4.
0
 x(7 x
0

4 x cos(3x 2 )dx
0
 /2
1
5.
ex
0 e x  7 dx
2
 1) dx
1/3
6.
 sin

/4
5
x cos xdx
 /6

7.
tan  2x  dx

x3
dx
9.  2
( x  6 x  1)3
0
e
8.
(ln x) 4
1 x dx
 /4
2
10.
2
 x x  9dx
0
2
x sec 2 xdx
0
4
11.
 tan
1
12.
cos x
dx
x
0

2
e7
5. 45/56 6. 7/48 7. 1/2ln1/2 8. 1/5 9. 72/289 10. 1/3 11. 98/3 12.
 3. 1/2sin1 4.
3
 8 
1. 7/480 2. ln 
2sin1
Warm Up: Name your u
Name:______________________________________
For the following problems, name the u you would use to solve by u-substitution.
1.

2.
 4xe
3.

4.
 tan
5.
 cos
6.

7.

8.
 x ln x dx
u=_____________
9.  sec 2 xe 4 tan x dx
u=_____________
10.  3 x sec 2 ( x 2 )dx
u=_____________
11.
3
sin x cos xdx
3 x2
u=_____________
dx
52 x
dx
x
2
3
u=_____________
u=_____________
x sec 2 xdx
u=_____________
x sin xdx
u=_____________
csc x csc x cot xdx u=_____________
ln x
dx
x
u=_____________
3

x2  1
x3  3x
dx
u=_____________
12.  cos 2t sin 5 2tdt
u=_____________
xe
13. 
dx
u=_____________
14. 
sin(1/ x)
dx
3x 2
u=_____________

e 2 y 1
dy
2 y 1
u=_____________
3 x4
15.
U-Substitution Practice #5
Name:____________________ Period:___
2 x2  x
 (4 x3  3x2 )2 dx
1.
2
 x x  4dx
2.
3.
 x( x  1)
4.  sin 5 x cos xdx
5.
 tan 9xdx
1/4
dx
6.
cos 2 x
 (1  sin 2 x)
2
dx
7.  sec 2 x(4 tan 3 x  3 tan 2 x)dx
8.
(ln x) 4
 x dx
 /4
x3
9.  2
dx
( x  6 x  1)3
0
2
10.
17
11.  tan( )d
2
12.
x sec 2 xdx
 ( x  9)
2/3
dx
10
0
2
3/2
1. 13 ( x  4)  C
2
0
1

 tan
3
2 1
2.  16 (4 x  3 x )  C
6
4. 16 sin x  C 5.  19 ln cos9x  C
5
8. 15 (ln x)  C 9. 72/289
10. 1/3
3.
4
9
( x  1)9/4  54 ( x  1)5/4  C
1
4
3
6.  12 (1  sin 2 x)  C 7. tan x  tan x  C
11. 12 ln(sec1) 12. 3
Worksheet #6: Initial Conditions
Name:___________________________________________Period:____
Solve the differential equation with the given initial condition.
1.
dy
 cos 2 x y ( 2 )  3
dx
2.
dy
 x3
dx
3.
dy
 8 x3  3 x 2  3 y (1)  1
dx
4.
dy
 sec 2 (3 x) y ( 4 )  2
dx
5.
dy
 e5 x
dx
6.
dy
 t
dt
y (0)  3
y (1)  1
y (1)  1
First, find f '( x ) . Then find f ( x ) .
7. f ''( x)  x3  2 x  1, f '(0)  1, f (0)  2
8. f ''( )  2cos  , f '( 2 )  1, f (0)  6
9. A particle located at the origin at t=1 begins moving
2
along the x-axis with velocity v(t )  12 t  t ft / sec . Let
s(t) be its position at time t. Find s(t).
10. A particle moves along the x-axis with velocity
v(t )  5t  t 2 ft / sec . Find s(t) assuming the particle is
located at x=5 at time t=2.
11. A particle located at the origin at time t=0 moves in a straight line with the acceleration a(t )  4  3t ft / sec2 .
A. State the solve the differential equation for v(t) assuming that the particle is at rest at time t=2.
B. Find s(t).
12. The velocity of a particle is v(t )  t 3  10t 2  24t ft / s Use your calculator to sketch a graph of the velocity.
Place units on your answers.
a. Find the equation for the position of the particle.
b. Using your calculator or your answer above find the displacement over [4,6] and [0,4]
c. Using your above answers, what is the particles total displacement from [0,6]?
d. What is the total distance traveled over [0,6]?
5
e. What is the meaning of
 v(t )dt ? What is the meaning of
0
5
 v(t ) dt ?
0
5x
4
3/2
1. y  12 sin(2 x)  3 2. y  14 x  34 3. y  2 x 4  x3  3x  1 4. y  13 tan(3x)  73 5. y  15 e  145 6. y  23 t  13
4
2
7. f '( x)  14 x  x  x  1 f ( x) 
1
20
x 5  13 x 3  12 x 2  1x  2 8. f '( )  2sin   1 f ( )  2 cos   x  8
3
2
2
3
2
3
2
9. s (t )  16 t  12 t  13 10. s (t )  52 t  13 t  73 11a. v(t )  4t  32 t  2 11b. s (t )  2t  12 t  2t
12a. s (t )  14 t  103 t  12t  C 12b. -20/3 ft, 128/3 ft 12c. 36 ft. 12d. 148/3 ft 12e. Total displacement of the
particle for the first 5 seconds. Total distance of the particle for the first 5 seconds.
4
3
2
Calculus
Name
Differential Equations
Find the general solution of the differential equations. Do work on your own paper.
1.
dy
 x3  5
dx
2.
dy
1
 8x 
dx
2
3.
dW
4 t
dt
4.
y  2   10
dr
 3sin p
dp
Solve the initial value differential equations.
5.
dy
 1  cos x
dx
y π   0
6.
dy
 6 x2  4 x
dx
7.
dP
 10e t
dt
P  0  25
8.
ds
 32t  100
dt
9.
dq
 2  sin z
dz
q (0)  5
10. y  e x 3  2 x  1
11. y 
x
9  x2
y(0) = -2
12.
dy
 2e t
dt
s  50
t 0
y  3  4
y(ln 2) = 0
13.
d2y
 sin x
dx 2
y (0)  3;
y(0)  0
14.
d2y
 2  6x
dx 2
y (0)  1;
y(0)  4
d3y 1

15.
dx3 t 3
y (1)  1;
The velocity v 
ds
dv
or acceleration a 
of a body moving along a coordinate line is given. Find the body’s
dt
dt
y(1)  3;
y(1)  2
position s at time t.
16. v  t   9.8t  5
18. a  t   cos t
s(0)  10
17. a(t )  32
s(0)  0; v(0)  20
s(0)  1; v(0)  1
On earth acceleration due to gravity is a(t) = -32 ft/sec2 or a(t) = -9.8 m/sec2.
19. A tomato is thrown upward from a bridge 25 m above the ground at 40 m/sec.
a. What is v(0)? What is s(0)?
b. Give formulas for the acceleration, velocity, and height (position) of the tomato at time t.
c. How high does the tomato go, and when does it reach its highest point?
velocity at the highest point? Find the value of t when this occurs.)
(Hint: What is
d. How long is the tomato in the air? (Hint: When is position 0?)
20. When Apollo 15 astronaut David Scott dropped a hammer and a feather on the moon to demonstrate that in a
vacuum all bodies fall with the same (constant) acceleration, he dropped them from about 4 ft. above the
ground. The television footage of the event shows the hammer and feather falling more slowly than on Earth,
where, in a vacuum, they would have taken only half a second to fall 4 ft. On the moon acceleration due to
gravity a(t)= 5.2 ft / sec2 .
a. What is the initial velocity and initial position?
b. Find the position equation s(t).
c. How long did it take the hammer and feather to fall 4 feet on the moon?
Answers: 1. y 
1
x4
8 3
 5 x  c 2. y  4 x 2  x  c 3. w  t 2  c
2
4
3
4. r  3cos p  c
20e 32
t  25 8. S  16t 2  100t  50
5. y  x  sin x  π 6. y  2 x  2 x  14 y 7. P 
3
3
2
9. q  2 z  cos z  4 10. y  e x 3  x 2  x  3
13. y   cos x  1, y   sin x  x  3
15. y  
1 5
 ,
2t 2 2
17. v  t   32t  20,
y 
1 5t
 ,
2t 2
2
11. y  9  x  5 12. y  2et  4
14. y  2 x  3x 2  4, y   x3  x 2  4 x  1
1
5
1
2
y  ln t  t 2 
16. v(t )  4.9t  5, s(t )  4.9t  5t  10
2
4
4
s t  16t 2  20t 18. v(t )  sin t  1, s(t )   cos(t )  t
2
19. a. 40, 25 b. v(0)  9.8t  40, s(t )  4.9t  40t  25 c.106.633m
20. a. 0, 4 b. s(t )  2.6t 2  4 c. 1.240 sec
d.
80
sec
9.8
More Practice Integration
Name: _________________________________
Answers are on the blog.
Find the indefinite integral.
1. 
3x (x
2. 
x 5  x 4  2x  1
dx
x3
2
 3) 4 dx
sin (3 )cos(3 )d 
3  2t d 
5. 
6. 
9.

dx
x ln x
10.

ex
dx
ex  4
13.
 5 sec(2x ) tan(2x )dx
16.

x 1
dx
x 2  2x
17.

11.
1  e 2x
dx
4.
2

8.
 cos(3 )d 

dx
18.

ln x 
4
dx
x
12.

15.
 cos x cos(sin x )dx
e 3x e 3x  3 dx
 sin(3x )  cos(2x )dx
14.
ex
2x
5 x 2 1
sec (7 x )dx
7. 
5
2
3. 
cos x
dx
sin x
 1
2
 3   dx
x
x
Evaluate the definite integral.

2
5
1
19.  1  cos(2 x)  dx 20.   3x  4 dx 21.  dx 22.
x
0
2
2

8
24.
25.
6
1
29.
 xe
0
9
5
x
  x  9e  dx
2
  dx
26.
3
 x2
 /3
dx 30.

0
sin 
d
cos 2 
8
4 x3 dx
e4
31.

e
dw
32.
w ln w

0
 /4
1
27.
1 u
du
u
0
4
3
  4sec x tan x  dx 23.
2
2
3 5
 x (1  2 x ) dx
0
1/2

0
sin 1 x
1  x2
dx
28.
 sin(4 x)dx
0
AP Calculus
Name___________________
Fundamental Theorem Worksheet
Evaluate the following:
x
4x
d
(3t 2  2t )dt
1.

dx 2
d
(3t  2)dt
2.
dx 2
x2
d
3.
cos tdt
dx 2
x3
d
3t  1
5.
( 2
)dt

dx 2 x 5t  2
5
4.
d
ln(t ) dt
dx 3x2
x3
d
6.
(4t 3  2t 2  3)dt

dx 2 x2
Evaluate the following:

10
7.
 (3x  4 x  8)dx
2
3
8.
 2
2
8
9.
1
 x  3 dx
 cos( x)dx
10.
 4x
12.
x
3
 3x 2  2 x  4
4
11.
1
x
6
dx
2
3

 4 x  5  4 dx
1
AP Calculus: Practice FTC/Basic Integrals
Name:_______________________ Period:___
Find the following using the Fundamental Theorem of Calculus.
x
x
d 2
t dt
1.
dx 2
d
4.
dx
2.
2
x
1
x 1  t 2 dt
d
t4
5.
 1  t dt
dx 1/3
5
d
8.
cos t dt
dx ex
d
sec t dt
7.
dx ex
d
10.
dx
sin 1 x

x
x4
ln x
t
e dt
e
d
cos tdt
dx 
d
1
dt
11.

dx sec x x
3.
d
tan 1 t dt
dx x
x2
d
1
6.
dt

dx  2 1  t 2
d
9.
dx
tan 1 x

t 3 dt
sin x
d
12.
dx
3
x
x
1/3
2x
 5 x dt
Simplifying the following:
13.
x
 5 dx
14.
 cos x dx
16.

x  5 x dx
17.

19.
 csc x cot x dx
20.
 1 x
22.

23.
e
4
1
1 x
2
dx
 /3
25.


26.
1
25 x dx
3/2

0
1
x
2
18.  csc 2 x dx
dx
dx


1
dx
1  x2
5
21.
x
24.

27.
1
x 1  x2
dx
/6
1
 20dx
1/ 3
1
1
1/2 1  x 2 dx

30.
2
32.
dx
 sin xdx
1
29.
2
 /3
sec 2 xdx
/2
15
31.
1
dx
x
 /3
csc x cot xdx
/6
28.
8
15.  sec x tan x dx
64
33.

16
1
dx
4
x
1
1  x2
dx
AP Calculus: Practice MC Basic Integration
1. (Calculator) Let F(x) be the antiderivative of
A. 0.048
B. 0.144
Name:__________________________________
 ln x 
3
x
C. 5.827
. If F(1)=0, then F(9)=
D. 23.308
E. 1,640.250
k
2. What are the values of k for which
 x dx  0
2
3
A. -3
B. 0
C. 3
D. -3 and 3
E. -3, 0, 3
2
B. e  e
e2
1
C.
e
2
2
2
D. e  2
e2 3
E.

2 2
B . 7/24
C. 1/2
D. 1
E. 2ln2
 x2 1 
1  x  dx
e
3. 
A. e 
2
4.
1
x
2
1
e
dx
1
A. -1/2
5.(Calculator) At time t  0 , the acceleration of a particle moving on the x-axis is a (t )  t  sin t . At t=0, the
velocity of the particle is -2. For what value of t will the velocity of the particle be zero?
A. 1.02
6.
B. 1.48
C. 1.85
D. 2.81
E. 3.14
1 2t
e dt 
2
t
A. e  C
b
7. If

a
B. e
 2t
C
t
C. e 2  C
t
D. 2e 2  C
E. et  C
b
f ( x)dx  a  2b , then  ( f ( x)  5)dx 
a
A. a+2b+5
B. 5b-5a
1.C 2. A 3. E 4. C 5. B 6. C 7. C
C. 7b-4a
D. 7b-5a
E. 7b-6a