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Laboratory: Differentiation Review and Antiderivatives
This lab is designed to be done as a group project with each group consisting of 3 to 5
students.
Part 1: Differentiation Review
1.
Read Part 2: Antiderivatives for instruction on how to make flash cards.
Use the table on the next three pages to create flash cards for the basic formulas for
differentiation. Fill any spaces left with basic derivatives that you should be able to
do without the use of pen and paper.
2.
The following problems should be done independently and not as a group.
Afterwards compare your answers with your group. Discuss and correct
discrepancies. The groups write-up should consist of a separate set of solutions
from each member of the group.
Evaluate and simplify the following derivatives:
(
1.
d
5x 3 + 8x + 7
dx
3.
d
4t 3 cos ( 5t )
dt
5.
d ⎛ 3z 2 + 4 ⎞
dz ⎜⎝ z 3 + z ⎟⎠
7.
d
dt
9.
d
x5 x
dx
11.
d 2
x e
dx
13.
d 2 −x
x e sin (π x )
dx
15.
d cos( x )
x
dx
(
(
)
( ))
t ln 5t 2
( )
(
x3 + 5
(
(
)
)
)
(
)
2.
d
3x x 3 + 5
dx
4.
d πx
e tan(x)
dx
6.
d ⎛ sin ( 3θ ) ⎞
dθ ⎜⎝ 4 + cos ( 3θ ) ⎟⎠
8.
d
sin −1 x 2
dx
10.
d ⎛ tan −1 s ⎞
ds ⎜⎝ s ⎟⎠
12.
d
ln sec 2 ( 3w )
dw
14.
d ⎛ y3 + 4 3 ⎞
⎜
⎟
dy ⎝ y + 1 ⎠
)
Page 75
(
(
)
( ))
( (
))
Part 2: Antiderivatives
The following 9 pages can be printed and used as flash cards. Cut along the solid lines
and fold on the dotted line. You can also cut on the dotted line and tape to 3 by 5 index
cards.
Make the flash cards and use them with your group to study and learn these formulas.
There are 51 formulas on these cards. It would be nice to have an even 60 formulas.
Make up 9 more antiderivative problems that you should be able to solve without the use
of pen and paper. Use the following table for your write-up.
52.
53.
54.
55.
56.
57.
58.
59.
60.
Page 80
1. ∫ x p dx,
2. ∫
p ≠ −1
1
dx
x
1. =
x p +1
+C
p +1
2. = ln x + C
1 kx
e +C
k
3. ∫ ekx dx
3. =
4. ∫ ln(x) dx
4. = x ln(x) − x + C
5. ∫ sin(kx) dx
1
5. = − cos(kx) + C
k
6. ∫ cos(kx) dx
6. =
Page 81
1
sin(kx) + C
k
1
ln sec(kx) + C
k
7. ∫ tan(kx) dx
7. =
8. ∫ cot(kx) dx
1
8. = − ln csc(kx) + C
k
9. ∫ sec(kx) dx
9. =
10. ∫ csc(kx) dx
1
10. = − ln csc(kx) + cot(kx) + C
k
11. ∫ sec 2 ( kx ) dx
11. =
12. ∫ csc 2 (kx) dx
1
12. = − cot(kx) + C
k
Page 82
1
ln sec(kx) + tan(kx) + C
k
1
tan(kx) + C
k
13. =
14. ∫ csc(kx)cot(kx) dx
1
14. = − csc(kx) + C
k
15. ∫
16. ∫
17. ∫
dx
,
2
x + k2
dx
k 2 − x2
dx
x x2 − k 2
18. ∫ e x dx
1
sec(kx) + C
k
13. ∫ sec(kx) tan(kx) dx
1 −1 ⎛ x ⎞
tan ⎜ ⎟ + C
⎝ k⎠
k
k>0
15. =
dx, k > 0
⎛ x⎞
16. = sin −1 ⎜ ⎟ + C
⎝ k⎠
dx,
k>0
17. =
1 −1 x
sec
+C
k
k
18. = e x + C
Page 83
19. ∫ sin(x) dx
19. = − cos(x) + C
20. ∫ cos(x) dx
20. = sin(x) + C
21. ∫ tan(x) dx
21. = ln sec(x) + C
22. ∫ cot(x) dx
22. = − ln csc(x) + C
23. ∫ sec(x) dx
23. = ln sec(x) + tan(x) + C
24. ∫ csc(x) dx
24. = − ln csc(x) + cot(x) + C
Page 84
25. ∫ sec 2 ( x ) dx
25. = tan(x) + C
26. ∫ csc 2 (x) dx
26. = − cot(x) + C
27. ∫ sec(x) tan(x) dx
27. = sec(x) + C
∫
28. csc(x)cot(x) dx
28. = − csc(x) + C
29. ∫
dx
x +1
29. = tan −1 ( x ) + C
30. ∫
dx
2
1− x
2
dx
30. = sin −1 ( x ) + C
Page 85
31. ∫
dx
x x −1
2
32. ∫ ( 3t + 5) dt
33. ∫
dz
z3
34. ∫ 5x −1 dx
dx
31. = sec −1 x + C
32. =
3 2
t + 5t + C
2
1
33. = ∫ z −3 dz = − z −2 + C
2
34. = 5 ln x + C
1
1
2
+1
35. ∫ x dx
x2
2 3
35. = ∫ x dx = 1
+ C = x2 + C
3
+1
2
36. ∫ t 7 dt
t8
36. = + C
8
Page 86
37. ∫ e− x dx
37. = −e− x + C
38. ∫ e 3θ dθ
38. =
39. ∫ sin(5θ ) dθ
1
39. = − cos(5θ ) + C
5
40. ∫ cos(4 β ) d β
40. =
1
sin(4 β ) + C
4
41. ∫ tan(6x) dx
41. =
1
ln sec(6x) + C
6
42. ∫ cot(π x) dx
42. = −
Page 87
1 3θ
e +C
3
1
ln csc(π x) + C
π
1
ln sec(πθ ) + tan(πθ ) + C
π
43. ∫ sec(πθ ) dθ
43. =
44. ∫ csc(3x) dx
1
44. = − ln csc(3x) + cot(3x) + C
3
45. ∫ sec 2 (π x ) dx
45. =
46. ∫ csc 2 (3t) dt
1
46. = − cot(3t) + C
3
47. ∫ sec(5α ) tan(5α ) dα
47. =
48. ∫ csc(2x)cot(2x) dx
1
48. = − csc(2x) + C
2
Page 88
1
tan(π x) + C
π
1
sec(5α ) + C
5
49. ∫
50. ∫
51. ∫
dx
x + 25
49. =
2
dx
4 − x2
dx
x x2 − 9
⎛ x⎞
50. = sin −1 ⎜ ⎟ + C
⎝ 2⎠
dx
dx,
1 −1 ⎛ x ⎞
tan ⎜ ⎟ + C
⎝ 5⎠
5
k>0
1
x
51. = sec −1 + C
3
3
Page 89
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