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

12.
Z
1
2 + 4 x 2
dx
13.
Z
4 x
(3 + x
2
)
2
dx
14.
Z
x
2
√
4 - x dx .
15.
Z
1
q
3
4
+ x - x
2
dx (HINT: Complete the square)
16.
Z
e
3 /x
x 2
dx
17.
Z
e
x
e 2 x + 1
dx
18.
Z
(sin 4 x )(cos 4 x )