Integral Calculus 201-NYB-05
Vincent Carrier
Exercise Sheet 2
1.2 Basic Formulas
Find an antiderivative for the following functions.
1. f (x) = x8
2. f (x) =
4. f (x) = − csc x cot x
5. f (x) =
7. f (x) =
1
1 + x2
√
4
x7
1
x ln 7
8. f (x) = sec x tan x
1
x
1
3. f (x) = √
5
x3
6. f (x) = ex
9. f (x) = 5x ln 5
12. f (x) = − sin x
10. f (x) = cos x
11. f (x) =
13. f (x) = − csc2 x
14. f (x) = sec2 x
1
15. f (x) = √
1 − x2
1.3 Basic Formulas Plus
Find an antiderivative for the following functions.
16. f (x) = e−5x
17. f (x) = − sin 3x
19. f (x) = sec2 (2 − 5x)
20. f (x) =
22. f (x) = sec(x/3) tan(x/3)
23. f (x) = − csc2 2x
25. f (x) =
1
1 + 4x2
√
28. f (x) = 2 3 3 − 2x
31. f (x) = x2 ex
3
34. f (x) = x3 sec2 x4
26. f (x) =
1
(2x + 3)4
1
2x
18. f (x) = (5 + 2x)6
21. f (x) = cos(−4x)
24. f (x) = e−x/3
27. f (x) = √
1
1 − 9x2
29. f (x) = 4 sin 3x
30. f (x) =
5e−2x
4
3
32. f (x) = √
2x + 3
33. f (x) =
x
1 + x2
e3x + x2
e3x + x3
36. f (x) =
x3
2 + x4
35. f (x) =
Answers:
1. F (x) =
x9
9
2. F (x) =
4x11/4
11
3. F (x) =
5x2/5
2
4. F (x) = csc x
5. F (x) = log7 |x|
6. F (x) = ex
7. F (x) = arctan x
8. F (x) = sec x
9. F (x) = 5x
10. F (x) = sin x
11. F (x) = ln |x|
12. F (x) = cos x
13. F (x) = cot x
14. F (x) = tan x
15. F (x) = arcsin x
16. F (x) =
e−5x
−5
17. F (x) =
19. F (x) =
tan(2 − 5x)
−5
20. F (x) = −
cos 3x
3
1
6(2x + 3)3
18. F (x) =
(5 + 2x)7
14
21. F (x) =
sin(−4x)
−4
22. F (x) = 3 sec(x/3)
23. F (x) =
cot 2x
2
24. F (x) = −3e−x/3
arctan 2x
2
26. F (x) =
ln |x|
2
27. F (x) =
25. F (x) =
28. F (x) = −
3(3 − 2x)4/3
4
3
29. F (x) = −
4 cos 3x
3
31. F (x) =
ex
3
√
32. F (x) = 3 2x + 3
34. F (x) =
tan x4
4
35. F (x) =
ln |e3x + x3 |
3
arcsin 3x
3
30. F (x) = −
5e−2x
8
33. F (x) =
ln(1 + x2 )
2
36. F (x) =
ln(2 + x4 )
4