Homework5.3 Ans - Integration by Substitution SUGGESTED...

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
1
4
( x
3
+ 3)
4
+ C
2.
Z
5
5 x + 3
dx
ln | 5 x + 3 | + C
3.
Z
2 x cos ( x
2
) dx
sin ( x
2
) + C
4.
Z
4 x ( x
2
+ 6)
2
dx
2
3
( x
2
+ 6)
3
+ C
5.
Z
sec (4 x ) tan (4 x ) dx
1
4
sec (4 x ) + C
6.
Z
(3 x - 5)
9
dx
1
30
(3 x - 5)
10
+ C
1

7.
Z
e
- 2 x
dx
-
1
2
e
- 2 x
+ C
8.
Z
sin x cos x
1 + sin
2
x
dx
1
2
ln (1 + sin
2
x ) + C
9.
Z
csc
x
2
cot
x
2
dx
- 2 csc
x
2
+ C
10.
Z
- 3 x
3
√
1 - x
4
dx
1
2
(1 - x
4
)
3
2
+ C ; Video Solution:
11.
Z
e
√
x
√
x
-
1
4
cos 4 x dx
2 e
√
x
-
1
16
sin (4 x ) + C
12.
Z
1
2 + 4 x
2
dx
1
2
√
2
arctan (
√
2 x ) + C ; Video Solution: