CALCULUS
WORKSHEET ON 5.6, INVERSE TRIG FUNCTIONS AND DIFFERENTIATION
Work the following on notebook paper. No calculator.
Evaluate.
1. arcsin
2. cos 1
1
2



1
2



6. arctan  
3. arccos  



4. sin 1  
9. arctan   1
5. tan 1 3
3

2 
3

2 
3

3 
10. arcsec   2 

7. arccos   1
11. csc1  2
8. sin 1  1
12. arc cot 


3

__________________________________________________________________________________
Evaluate.
3

4


 4 
14. sec  sin 1    
 5 


 5 
15. csc  arctan    
 12  


 2 
16. tan  cos 1    
 3 

13. sin  arctan 



 x 
 3 
17. cos sin 1  2x 
18. tan  arcsec   


19. sin arctan  5x 

_________________________________________________________________________________
Find the derivative.
20. f  x   arcsin  3x 
 
 
24. y  arctan e x

25. f  x   sin arccos  2 x 
21. y  cos 1 5 x 2
 
22. g  x   2arcsec x3
 

26. g  x   x arcsin x3
 x
3
23. h  x   tan 1  
__________________________________________________________________________________
1
27. Write an equation for the tangent line to the graph of y  2arcsin x at the point where x  .
2
 x
2
28. Write an equation for the tangent line to the graph of y  tan 1   at the point where x = 2.
29. A fish is reeled in at a rate of 2 ft/sec from a bridge 16 ft above the water. At what rate is the
angle between the line and the water changing when there are 20 ft of line out? Use inverse
trig functions to find the answer.
Answers to Worksheet on 5.6, Inverse Trig Functions and Differentiation
1.

6
4. 

3
7. 
2
3
13.
3
5
16. 
11. 
5
2
5.

3
6. 
25 x  1
6
x
25. 
27. y 
28, y 
x6  1
20.
23.
4x
1 4x
2

4 
1
x 
2
3
3

4


2 rad
15 sec
1
 x  2
4

2

4
12.
1  4x 2
3
1  9x
21. 
2
3
9  x2
26.
24.
3x3
1 x
6
 arcsin  x3 

6
9. 

4
15. 
13
5
18.
x2  9
3
5
6
5
3
17.
2
22.
29.
3.
14.
5x
19.

3
8. 
10.
5
6
2.
10 x
1  25 x 4
ex
1  e2 x
CALCULUS BC
WORKSHEET ON INVERSE TRIG FUNCTIONS AND REVIEW
Work the following on notebook paper.
Find the derivative.
2. y  arccos 2 x3
arcsin  2x 
x
2
6. h  x   x arctan x 2
3. g  x   arcsec  3x 
7. p  x   cos  arcsin x 
 x
5
8. q  x   sec  arctan x 
1. f  x   3arcsin  5 x 
5. y 
 
 
4. f  x   arctan  
____________________________________________________________________________________
Evaluate.
dx

 25  x 2
dx
10. 

 4  x2
13. 

dx
 x  6 x  34
9. 
2
dx

 8  2 x  x2
2x  7
dx
15. 
 2
 x  4 x  13
14. 
dx

 x x2  9
 x3
dx
12. 
 16  x 2
11. 
3  2x

dx
 10 x  x 2  9
16. 
_________________________________________________________________________________________
5
, find  f 1   4  .
6
 1
18. Let f  x   cos x, 0  x   . Find  f 1     .
 2
17. If f  3  4 and f   3 
__________________________________________________________________________________________
19. (2005 Form B – AB 4) (No calc)
The graph of the function f shown on the right consists of three line segments.
(a) Let g be the function given by g  x   
x
4
f  t  dt. For each of
g   1 , g   1 , and g   1 , find the value or state that it does
not exist.
(b) For the function g defined in part (a), find the x-coordinate of each
point of inflection of the graph of g on the open interval  4  x  3.
Explain your reasoning.
(c) Let h be the function given by h  x    f  t  dt. Find all values
3
x
of x in the closed interval  4  x  3 for which h  x   0.
(d) For the function h defined in part (c), find all intervals on which h is decreasing. Explain your
reasoning.
Answers to Worksheet on Inverse Trig Functions and Review
2. 
3.
4.
5.
 x 1 
C
 3 
15
1.
1  25x
6x2
14. arcsin 
2
 x2
C
 3 
 x 5
16. 2 10 x  x 2  9  7 arcsin 
C
 4 
15. ln x 2  4 x  13  arctan 
1  4 x6
1
x 9x2 1
5
25  x 2
17.
2 x  arcsin  2 x  1  4 x 2
x
3
2
1 4x
2
 
2x
 2 x arctan x 2
1  x4
x
7. 
1  x2
x
6.
8.
1  x2
 x
9. arcsin    C
5
1
 x
10. arctan    C
2
2
 x
1
11. arcsec    C
3
3
x
4
12.  16  x 2  3arcsin    C
13.
1
 x 3
arctan 
C
5
 5 
6
5
18. 
2
3
19. See AP Central
CALCULUS BC
WORKSHEET ON 8.1
Work the following on notebook paper. No calculator.
Evaluate the given integrals.
3
2
 x 5
dx 
8. 
0 x  2
2


1.   9 x  3  3sec x tan x  7sec2 x  dx
x


x4

2. 
dx
2
 x  8x  1
 
3.
x
4.
5
 sin  3x  cos  3x  dx
5.
2
1 x  x  1 dx 
3
cos 5 x 4 dx

10.

9
e2
x
x
 
cos e x dx 
3 x4
7 dx 
6
dx
 10 x  x 2
1

12. 
dx
2
 x  4x  9
2x  7
dx
13. 
 2
 x  4 x  13
sin  3x  dx 
 ln x 4
x

12

7. 
e
0
 1 e
11. 
3
2
6. 
9.
 x3
dx
 16  x 2
dx 
14. 
__________________________________________________________________________________________
Multiple Choice. All work must be shown.
15.
Which of the following represents the area of the shaded region in the figure above?
 c f  y  dy
(D)  b  a   f  b   f  a 
(A)
d
 a  d  f  x   dx
(E)  d  c   f  b   f  a 
(B)
b
(C) f   b   f   a 
_________________________________________________________________________________________
dy

16. If x3  3xy  2 y 3  17, then in terms of x and y,
dx
x2  y
x2  y
x2  y
x2  y
 x2



(A) 
(B)
(C)
(D)
(E)
x  2 y2
x  y2
x  2y
2 y2
1  2 y2
TURN->>>
 3x 2
17. 
 x3  1
dx 
(A) 2 x3  1  C
(B)
3
x3  1  C
2
(C)
x3  1  C
(D) ln x3  1  C


(E) ln x3  1  C
_________________________________________________________________________________________
18. For what value of x does the function f  x    x  2  x  3 have a relative maximum?
2
(A)  3
(B) 
7
3
(C) 
5
2
(D)
7
3
(E)
5
2
Answers to Worksheet on 8.1
9 x2
1
1.
 2  3sec x  7 tan x  C
2
x
2.
x2  8x  1  C
3.
1
sin 5 x 4  C
20
4.
5.
6.
7.
8.
9.
 
sin 6  3x 
C
18
609
8
1 1
2
  

3 2
2 
31
5
3
2
  ln
2
5
1
sin1  sin
e
4
7x
10.
C
4ln 7
 x 5
11. 6arcsin 
C
 5 
1
 x2
12.
arctan 
C
5
 5 
 x2
13. ln x 2  4 x  13  arctan 
C
 3 
x
14.  16  x 2  3arcsin    C
4
15. B
16. A
17. A
18. D
CALCULUS BC
WORKSHEET ON INTEGRATION BY PARTS
Work these on notebook paper. No calculator.
Evaluate.
 xe dx
2.  x sec2 x dx
2x
1.
 x sin x dx
4.  x3 ln x dx
x
5.  3x dx
e
6.  arctan  2x  dx
3.
2
7.
8.
 e sin x dx
5x
 xe dx
4x
1
0
e
 e x ln x dx
10.  arcsin  3x  dx
11.  x3e2x dx
12.  ln  x 2  1 dx
9.
________________________________________________________________________________________
13. (2003 AB 5) (No calc)
A coffeepot has the shape of a cylinder with radius 5 inches, as shown in the figure.
Let h be the depth of the coffee in the pot, measured in inches, where h is a function
of time t, measured in seconds. The volume V of coffee in the pot is changing at a
rate of  5 h cubic inches per second. (The volume V of a cylinder with radius r
and height h is V   r 2h.)
(a) Show that
dh
h

.
dt
5
(b) Given that h = 17 at time t = 0, solve the differential equation
for h as a function of t.
(c) At what time t is the coffeepot empty?
dh
h

dt
5
Answers to Worksheet on Integration by Parts
1
1
1. xe2 x  e2 x  C
2
4
2. x tan x  ln cos x  C
3.  x 2 cos x  2 x sin x  2cos x  C
4.
5.
6.
7.
8.
9.
x 4 ln x x 4

C
4
16
1
1
 xe 3 x  e 3 x  C
3
9
1
x arctan  2 x   ln 1  4 x 2  C
4
1
4
 e4 x cos x  e4 x sin x  C
17
17
6 5 1
 e 
25
25
e2
4


1  9 x2
C
3
1
3
3
3
11. x 3e2 x  x 2e2 x  xe2 x  e2 x  C
2
4
4
8
2
12. x ln x  1  2 x  2arctan x  C
10. x arcsin  3x  


13. See AP Central
CALCULUS
WORKSHEET ON 8.1 – 8.2
Work the following on notebook paper. No calculator.
2x
1. 
dx

 x4
x 1

dx
 x2  2 x  4
2. 
1

dx
 2  2 x  x2
8. 
9.
 arctan 3x  dx
1
3.
 3x
 xe dx
10.
4.
 sec  4x  dx
11. 

ln x
5. 
 2 dx
 x
 sin x
6. 
dx
 cos x
7.

0
x sin  2 x  dx
0 e
x
sin x dx
2x  5
dx
 x  2x  2
12.
2
 arcsin 5x  dx
3
 x dx
13.  2
 x 4
14.

1 2 x
x e dx
0
_________________________________________________________________________________________
Multiple Choice. All work must be shown.

3
x
15. If f  x   sin   , then there exists a number c in the interval
that satisfies the
x
2
2
2
conclusion of the Mean Value Theorem. Which of the following could be c?
2
3
5
3
(A)
(B)
(C)
(D) 
(E)
3
4
6
2
_________________________________________________________________________________________
2
16. If f  x    x  1 sin x, then f   0  
(A)  2
(B)  1
(C) 0
(D) 1
(E) 2
_________________________________________________________________________________________
17. The acceleration of a particle moving along the x-axis at time t is given by a  t   6t  2. If the
velocity is 25 when t = 3 and the position is 10 when t = 1, then the position x  t  
(A) 9t 2  1
(B) 3t 2  2t  4
(C) t 3  t 2  4t  6
(D) t 3  t 2  9t  20 (E) 36t 3  4t 2  77t  55
_________________________________________________________________________________________
d x
18.
cos  2 u  du is
dx  0
1
1
(A) 0
(B)
(C)
(D) cos  2 x 
(E) 2 cos  2 x 
sin x
cos  2 x 
2
2
Answers to Worksheet on 8.1 – 8.2
1. 2 x  8ln x  4  C
2.
3.
4.
5.
6.
x2  2 x  4  C
1
1
 xe 3 x  e 3 x  C
3
9
1
ln sec  4 x   tan  4 x   C
4
1
1
 ln x   C
x
x
 2 cos x  C
7. 

2
 x 1
8. arcsin 
C
 3 
1
9. x arctan  3x   ln 1  9 x 2  C
6
e sin1  e cos1  1
10.
2
2
11. ln x  2 x  2  7arctan  x  1  C

12. x arcsin  5x  
1
1  25x 2  C
5
x2
 2ln x 2  4  C
2
14. e  2
13.
15. D
16. D
17. C
18. D



CALCULUS BC
WORKSHEET 1 ON 8.1 – 8.3
Work the following on notebook paper. No calculator.
1.
3
2
 cos  2 x  sin  2 x  dx
7.
4
 cos  6x  dx
2
2x 1
dx
 x  6 x  25
8.
x
sin  3x  dx
9.
e
2. 

3.
x
4.
 arcsin  4x  dx
10. 
5.
2
 sin  5x  dx
11.
3
2
 x  3x dx
 x2  1
12.
2
2
ln x dx
2x
sin x dx
dx

 12  4 x  x 2

0

6. 
0
2
2
cos3 x dx
x cos x dx
__________________________________________________________________________________________
Multiple Choice. All work must be shown.
x
f  x
2
10
5
30
7
40
8
20
13. The function f is continuous on the closed interval [2, 8] and has values that are given in the table above.
Using the subintervals [2, 5], [5, 7], and [7, 8], what is the trapezoidal approximation of
(A) 110
(B) 130
(C) 160
(D) 190
 2 f  x  dx ?
8
(E) 210
_________________________________________________________________________________________
14. What is the minimum value of f  x   x ln x ?
(A)  e
(B)  1
(C) 
1
e
(D) 0
(E) f  x  has no minimum value.
_________________________________________________________________________________________
1
1
15. At what value of x does the graph of y  2  3 have a point of inflection?
x
x
(A) 0
(B) 1
(C) 2
(D) 3
(E) At no value of x
Answers to Worksheet 1 on 8.1 – 8.3
1
1
1. sin 3  2 x   sin 5  2 x   C
6
10
5
 x  3
2. ln x 2  6 x  25  arctan 
C
4
 4 
1
2
2
3.  x 2 cos  3x   x sin  3x   cos  3x   C
3
9
27
1
2
4. x arcsin  4 x  
1  16 x  C
4
1
1
5. x 
sin 10 x   C
2
20
x2
1
6.
 3x  ln x 2  1  3arctan x  C
2
2
3
1
1
7. x  sin 12 x  
sin  24 x   C
8
24
192
x3
x3
8.
ln x 
C
3
9
1
2
9.  e2 x cos x  e2 x sin x  C
5
5
 x2
10. arcsin 
C
 4 
2
11.
3

12.

2
13. C
14. C
15. C
1

CALCULUS BC
WORKSHEET 2 ON 8.1 – 8.3
Work the following on notebook paper. No calculator.
1.
 sec  4 x  tan  4 x  dx
7.
 sin 3x  cos  2 x  dx
2.
 tan 3x  sec 3x  dx
8.
 sec  7 x  tan  7 x  dx
3.
 cos  2 x  sin  2 x  dx
9.
 sin 5x  cos 5x  dx
6
5
3
2
2
3
2
2
2x  3
dx
 x  10 x  41
10.
5.
2
 x sin 3x  dx
11.
0
6.
 arcsin 3x  dx
12.
 cos 5x  cos  4 x  dx
4. 

2
e
3x

6
cos x dx
x cos  2 x  dx
_________________________________________________________________________________________
13. (2004 Form B - AB 2) (Calc )
For 0  t  31, the rate of change of the number of mosquitoes on Tropical Island at time t days is
t
modeled by R  t   5 t cos   mosquitoes per day. There are 1000 mosquitoes on Tropical Island at
 5
time t = 0.
(a) Show that the number of mosquitoes is increasing at time t = 6.
(b) At time t = 6, is the number of mosquitoes increasing at an increasing rate, or is the number of mosquitoes
increasing at a decreasing rate? Give a reason for your answer.
(c) According to the model, how many mosquitoes will be on the island at time t = 31? Round your answer to
the nearest whole number.
(d) To the nearest whole number, what is the maximum number of mosquitoes for 0  t  31? Show the
analysis that leads to your conclusion.
Answers to Worksheet 2 on 8.1 – 8.3
sec6  4 x 
1 2
1
1
1.
 C or
tan  4 x   tan 4  4 x   tan 6  4 x   C
24
8
8
24
6
tan  3x 
1
1
1
2.
 C or
sec6  3x   sec4  3x   sec2  3x   C
18
18
6
6
sin 3  2 x  sin5  2 x 
3.

C
6
10
7
 x 5
4. ln x 2  10 x  41  arctan 
C
4
 4 
x2
2x
2
5.  cos  3x   sin  3x   cos  3x   C
3
9
27
1
6. x arcsin  3x   1  9 x 2  C
3
1
1
7.  cos x  cos  5 x   C
2
10
sec3  7 x 
8.
C
21
1
1
9. x 
sin  20 x   C
8
160
1 3x
3
10.
e sin x  e3 x cos x  C
10
10
 33
11.
24
1
1
12. sin x  sin  9 x   C
2
18
13. See AP Central
CALCULUS BC
WORKSHEET ON 8.1 – 8.4
Work the following on notebook paper. No calculator.

1
1. 

  25  x

1
2. 
 1  x
3.
2

2 2
2
3
dx
4.
2
cos  4 x  dx


 x
2
7.
 x
2
5.  tan 3   sec3   dx
dx
 cos 5x  sin 5x  dx
3
x
2
6.
2
0
16  4x 2 dx
 sin  4x  dx
2
1

dx
2
 x 9
8. 
9.
 cos 3x  cos  2 x  dx
__________________________________________________________________________________________
10. Given the region bounded by the graphs of y  ln x, y  0, and x  e . Find
(a) the area of the region
(b) the volume of the solid generated by revolving the region about the x-axis.
__________________________________________________________________________________________
Multiple Choice. All work must be shown.
11. If y  xy  x 2  1 , then when x  1,
1
dy
is
dx
1
(C)  1
(D)  2
(E) nonexistent
2
2
__________________________________________________________________________________________
(A)
(B) 
12. Let f be a function defined for all real numbers x. If f   x  
4  x2
x2
, then f is decreasing on
the interval
(A)   , 2 
(B)   ,  
(C)   2, 4 
(D)   2,  
(E)  2,  
__________________________________________________________________________________________
 x3 for x  0
. Which of the following statements
 x for x  0
13. Let f be the function defined by f  x   
about f is true?
(A) f is an odd function.
(B) f is discontinuous at x = 0.
(C) f has a relative maximum.
(D) f   0   0 .
(E) f   x   0 for x  0.
__________________________________________________________________________________________
14.
The graph of f  , the derivative of f is shown in the figure above. Which of the following describes all
relative extrema of f on the open interval (a, b)?
(A) One relative maximum and two relative minima
(B) Two relative maxima and one relative minimum
(C) Three relative maxima and one relative minimum
(D) One relative maximum and three relative minima
(E) Three relative maxima and two relative minima
Answers to Worksheet 1 on 8.1 – 8.4
x
1.
C
25 25  x 2
1
x
2. arctan x 
C
2
2 1  x2

3.
4.
5.
6.
7.

sin  5 x  sin  5 x 

C
15
25
x2
x
1
sin  4 x   cos  4 x   sin  4 x   C
4
8
32
 x
 x
2sec5   2sec3  
2
 
 2 C
5
3
2
1
1
x  sin 8 x   C
2
16
3
8. ln
x

3
5
x2  9
C
3
1
1
sin  5x   sin x  C
10
2
10. (a) 1
(b)   e  2 
9.
11. B
12. A
13. E
14. A
CALCULUS BC
WORKSHEET 2 ON 8.1 – 8.4
Evaluate. Do not use your calculator.
cos  4 x  dx
1.
x
2.
2
 sin 3x  dx
3.
4.
dx


2
x 2 25  x 2
3
0
2
arcsin x
1 x
2
dx
 arctan 5x  dx
5.
3
 cos  2x  dx
6.
x
7.
2x  3
 x2  8x  25 dx
11.

8.
x 2  9 dx
x
12.


5
ln x dx
9.
10.
2x  5

4 x  x2
dx
x3
4
x 2  16
0
1
0
dx
xe 2x dx
__________________________________________________________________________________________
Use your calculator, and give your answers correct to three decimal places.
13. (2003 Form B - AB 4)
A particle moves along the x-axis with velocity at time t  0 given by v  t   1  e1 t .
(a) Find the acceleration of the particle at time t = 3.
(b) Is the speed of the particle increasing at time t = 3? Give a reason for your answer.
(c) Find all values of t at which the particle changes direction. Justify your answer.
(d) Find the total distance traveled by the particle over the time interval 0  t  3 .
Answers to Worksheet 2 on 8.1 – 8.4
x2
x
1
1.
sin  4 x   cos  4 x   sin  4 x   C
4
8
32
1
1
2. x  sin  6 x   C
2
12
3. 
25  x 2
C
25 x
2
18
1
1
5. sin  2 x   sin 3  2 x   C
2
6
6
6
x ln x x
6.
 C
6
36
4.
5
 x4
7. ln x 2  8 x  25  arctan 
C
3
 3 
x
8. x 2  9  3arcsec    C
 3
1
9. x arctan  5x   ln 1  25 x 2  C
10
 x2
10. 2 4 x  x 2  arcsin 
C
 2 
64 2 128
11. 

3
3
3
1
12.  e  2 
4
4

13. See AP Central

CALCULUS BC
WORKSHEET ON 8.1 – 8.5
Work the following on notebook paper. No calculator.
5
dx
1. 
 2
2 x  1
4.
5 x
2. 
dx
 2
 2x  x 1
2
 7 x  16 x  5 dx
5.  3
 x  2 x2  x

3. 

0
3
2
x2
1  x 
2
3
 sin  6x  dx

1
6. 

dx
  x  3
2
7. arctan  5x  dx

3
2
3
2

x 1
8. 
dx
2
1 x  x  1
dx
9.
2
e
x
cos  2 x  dx
_________________________________________________________________________________________
 x
2
10. Given the region bounded by the graphs of y  cos   , y  0, x  0, and x   .
Find the volume of the solid generated by revolving the region about the x-axis.
________________________________________________________________________________________
Find the derivative.
11. f  x   arcsin  3x 
13. y  arctan e x
 
 

14. f  x   sin arccos  2 x 
12. y  cos 1 5 x 2

__________________________________________________________________________________________
Multiple Choice. All work must be shown.
1
15. An antiderivative for 2
is
x  2x  2
2
x2
(A)  x 2  2 x  2
(B) ln x 2  2 x  2
(C) ln
x 1




(D) arc sec  x  1
(E) arctan  x  1
_________________________________________________________________________________________
16. The region enclosed by the x-axis, the line x = 3, and the curve y 
is the volume of the solid generated?
(B) 3 3 
(A) 3
(C)
9
(D) 9

x is rotated about the x-axis. What
(E)
36 3

2
5
_________________________________________________________________________________________

17. 
0

(A)
3
3
dx
4  x2

(B)

4
(C)

6
(D)
1
2
ln 2
(E)  ln 2
Answers to Worksheet on 8.1 – 8.5
1
1. ln 2
2
3
2. ln 2 x  1  2ln x  1  C
2
3.
4.
5.
6.
7.
8.
9.
10.
3

3
1
1
 cos  6 x   cos3  6 x   C
6
18
4
5ln x  2ln x  1 
C
x 1
x
C
3 x2  3
1
x arctan  5x   ln 1  25 x 2  C
10
3
1

ln 2  ln5  arctan 2 
2
2
4
1 x
2 x
e cos  2 x   e sin  2 x   C
5
5

2
2
11.
12. 
3
1  9x 2
10 x
1  25 x 4
ex
1  e2 x
4x
14. 
1  4x2
13.
15. E
16. C
17. A

CALCULUS BC
MIXED INTEGRATION WORKSHEET
Work the following on notebook paper. No calculator.
1.
 x ln x dx
3

2.
0
4
5.
6. 

sin 3 x dx
4.
x
2
4
3

 x2  9
3. 
dx
x

sin  4 x  dx

9. 

 tan x sec x dx
3
4
3
3
3x 2  x  9
dx
x  x2  9
dx
x 4
10.
sin  3x  cos  2 x  dx

11. 
 13
2
e
x
sin  5 x  dx
3
7.
0
8.
 sin 5x  cos 5x  dx
2
2
3
dx
9 x2  1
2
__________________________________________________________________________________________
12. Solve for y: y  arctan  3x 
__________________________________________________________________________________________
Find the derivative.
13. f  x   arcsin x3
15. y  arctan  7 x 
 
14. y  cos 1 e x
 

16. f  x   cos arcsin  5 x 

__________________________________________________________________________________________
Multiple Choice. All work must be shown.
dy
17. If
 2 y 2 and if y   1 when x  1, then when x = 2, y =
dx
2
1
1
2
(A) 
(B) 
(C) 0
(D)
(E)
3
3
3
3
_________________________________________________________________________________________
18. The top of a 25-foot ladder is sliding down a vertical wall at a constant rate of 3 feet per minute.
When the top of the ladder is 7 feet from the ground, what is the rate of change of the distance
between the bottom of the ladder and the wall?
7
7
(A)  feet per minute
(B)  24 feet per minute
(C)
feet per minute
24
8
7
21
(D)
feet per minute
(E)
feet per minute
8
25
Answers to Mixed Integration Worksheet
x 4 ln x x 4
1.

C
4
16
2 5 2
2. 
3 12
x
3. x 2  9  3arcsec    C
 3
1
1
1
4.  x 2 cos  4 x   x sin  4 x   cos  4 x   C
4
8
32
sec6 x sec4 x
tan 4 x tan 6 x
5.

 C or

C
6
4
4
6
1 5
6. ln
4 3
3
7.
5
1
1
8. x 
sin  20 x   C
8
160
1  27  
9. ln   
2  4  36
1 x
5
10.
e sin  5x   e x cos  5x   C
26
26
1 2 3
11. ln 

3  2 1 
1
12. x arctan  3x   ln 1  9 x 2  C
6

13.
14, 
3x 2
1  x6
ex
1  e2 x
7
15.
1  49x 2
25 x
16. 
1  25 x 2
17. B
18. D
