Substitution With Definite Integrals
SUGGESTED REFERENCE MATERIAL:
As you work through the problems listed below, you should reference Chapter 5.9 of the recommended textbook (or the equivalent chapter in your alternative textbook/online resource) and your lecture notes.
EXPECTED SKILLS:
• Be able to evaluate definite integrals using a substitution of variables.
PRACTICE PROBLEMS:
For problems 1-3, use the given substitution to express the given integral (including the limits of integration) in terms of the variable u . Do not evaluate the integrals.
1.
Z
5
(3 x − 4)
10 dx , u = 3 x − 4
1
1 Z
11 u
10 du
3
− 1
2.
Z e
(ln x ) 3 dx , u = ln x
1 e x
Z
1
− 1 u
3 du ; Detailed Solution: Here
3.
Z
4
0
1
2 x + 1 dx , u = 2 x + 1
1 Z
9
2
1
1 u du
For problems 4-19, evaluate the following integrals.
4.
Z
1
2 x 3
√
1 − x 4 dx
0
4 −
8
√
15
; Detailed Solution: Here
1

5.
Z
4
π sin
2
(3 x ) cos (3 x ) dx
0
√
2
36
6.
ln 10
Z e
4 x dx
1
2500 −
1
4 e
4
7.
Z
2
π
(csc
2 x − sin x cos x ) dx
π
3
1
√
3
−
1
8
; Detailed Solution: Here
8.
Z
1
1 dx
1 + 3 x 2
− 1
2 π
√
3
; Detailed Solution: Here
9
9.
π
Z
12 sec
2
(4 x ) dx
0
√
3
4
10.
Z
10
1
2 + x dx
− 1 ln 12
11.
Z
2
(2 x + 2)( x
2
+ 2 x − 3) dx
− 3
25
2
2

12.
Z
6
π cos
4
(3 x ) sin (3 x ) dx
0
1
15
13.
Z
16
1
√ x e
√ x dx
1
− 2 e + 2 e
4
; Video Solution: http://www.youtube.com/watch?v=0plDXxeFNvQ
14.
Z
0
√
2 sin
− 1 x
√
1 − x 2 dx
π
2
32
15.
Z
4
π tan ( x ) dx
π
6
1
2
(ln 3 − ln 2); Detailed Solution: Here
16.
Z
2 x
2
( x 3 − 4) 3 dx
√
5
5
32
17.
Z
1
1 x 2 − 4 x + 4 dx
− 1
2
3
18.
Z
5 x
√ x − 4 dx
4
46
15
; Video Solution: https://www.youtube.com/watch?v=LE52v9SV po
3

19.
Z
6 p
3 + | x | dx
− 1
70
− 4
√
3
3
20. For each of the following, express the given definite integral (including the limits of integration) in terms of u . Then, evaluate the “new” integral by using an appropriate formula from geometry.
(a)
Z
0
π
√
2 x
3
√
4 − x 8 dx (HINT: Express x
8 as the square of some term).
4
(b)
Z e
4 p
16 − (ln x ) 2 dx x
1
4 π
8
21. It can be shown that
4 x 2 + 4 x − 15
=
1
2 x − 3
−
1
2 x + 5
.
(a) Let t be a fixed constant such that t > 2. Use these facts to evaluate
Z t
2
8
4 x 2 + 4 x − 15 dx .
1
2 ln
2 t − 3
2 t + 5
+ ln 3
(b) Evaluate lim t → + ∞
Z t
2
8
4 x 2 + 4 x − 15 dx ln 3
4