integration by substitution - Integration by Substitution...

Integration by Substitution
SUGGESTED REFERENCE MATERIAL:
As you work through the problems listed below, you should reference Chapter 5.3 of the rec-
ommended textbook (or the equivalent chapter in your alternative textbook/online resource)
and your lecture notes.
EXPECTED SKILLS:
• Know how to simplify a “complicated integral” to a known form by making an appro-
priate substitution of variables.
PRACTICE PROBLEMS:
For problems 1-21, evaluate the given indefinite integral and verify that your
answer is correct by differentiation.
1.
Z
3 x
2
( x
3
+ 3)
3
dx
2.
Z
5
5 x + 3
dx
3.
Z
2 x cos ( x
2
) dx
4.
Z
4 x ( x
2
+ 6)
2
dx
5.
Z
sec (4 x ) tan (4 x ) dx
6.
Z
(3 x - 5)
9
dx
7.
Z
e
- 2 x
dx
8.
Z
sin x cos x
1 + sin
2
x
dx
9.
Z
csc
x
2
cot
x
2
dx
10.
Z
- 3 x
3
√
1 - x
4
dx
11.
Z
e
√
x
√
x
-
1
4
cos 4 x dx
1

2 / 3
dx
19.
Z
csc
2
(3 x ) tan
2
(3 x ) + x
2
e
x
3
dx
20.
Z
1
x ln x
dx
21.
Z
cos - 1 x
√
1 - x 2
dx
Z
1
22. Use an appropriate trigonometric identity followed by a reasonable substitution to
evaluate
Z
tan x dx
2 + 77 x + 49 2 1
23. It can be shown that
32 x
(3 x + 1)(4 x + 5)
2
=
3 x + 1
-
(4 x + 5)
2
. Use this fact to evaluate
Z
32 x 2 + 77 x + 49
(3 x + 1)(4 x + 5) 2
dx .
24. Using the substitution x = sin θ for -
π
2
≤ θ ≤
π
2
, evaluate
Z
√
1 - x 2 dx . Express
your answer completely in terms of the variable x .
HINT - The following trigonometric identities will be helpful: sin
2
θ + cos
2
θ = 1,
cos 2 θ =
1
2
(1 + cos (2 θ ), and sin (2 θ ) = 2 sin θ cos θ
2