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Warm-up:. Evaluate the integrals. 1) 2). Warm-up:. Evaluate the integrals. 1) 2). Warm-up:. Evaluate the integrals. 1) 2). Integration by Substitution. Section 6.3. U-Substitution. Can be used to transform complicated integration problems into simpler ones. U-Substitution. - PowerPoint PPT Presentation
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Warm-up:Evaluate the integrals.
1)
2)
dx
xex
73
dxx
x )13
1(
2
Warm-up:Evaluate the integrals.
1)
2)
dx
xex
73
dxx
x )13
1(
2
Cxex ln73
Warm-up:Evaluate the integrals.
1)
2)
dx
xex
73
dxx
x )13
1(
2
Cxex ln73
Cxx
3
sin
3
2 12
3
Integration by Substitution
Section 6.3
U-Substitution• Can be used to transform complicated
integration problems into simpler ones.
U-Substitution• Can be used to transform complicated
integration problems into simpler ones.
* In general, there are no hard and fast rules for choosing u, and in some problems no choice of u will work. For those problems, we have other methods that will be discussed later. Choosing a u that is appropriate comes with practice and experience, but the guidelines will help to gain a better understanding.
Guidelines for Choosing U
1) Look for a substitution that would produce an integral expressed entirely in terms of u and du.
2) Evaluate the integral in terms of u.
3) Replace the u, so that your final answer is in terms of x.
Examples1) Evaluate
2) Evaluate
xdxx 21502
dxx 9sin
Examples1) Evaluate
2) Evaluate
xdxx 21502
dxx 9sin
12 xu
dxxdu )2(
dxx
du
2x
duxu2
250 Cu
51
51
duu 50
Cx
51
)1( 512
9xu
dxdu udusin
Cu cos
More Examples
3) Evaluate
4) Evaluate
xdx5cos
dx
x5
831
1
More Examples
3) Evaluate
4) Evaluate
xdx5cos
dx
x5
831
1
xu 5
dxdu 5
dxdu
5
5
cosduu uducos
5
1Cu sin
5
1
Cx 5sin5
1
83
1 xu
dxdu3
1
dxdu 3
duu3
15 duu 53
Cu
43
4
Cx
4)8
31(4
3
More difficult Examples5) Evaluate
6) Evaluate dxxx
2sec1
2251 x
dx
More difficult Examples5) Evaluate
6) Evaluate dxxx
2sec1
xu 5 2251 x
dx
More difficult Examples5) Evaluate
6) Evaluate dxxx
2sec1
xu 5dxdu 5
dxdu
5
2251 x
dx
51
12
du
u
215
1
u
duCu 1tan
5
1Cx 5tan
5
1 1
xu
More difficult Examples5) Evaluate
6) Evaluate dxxx
2sec1
xu 5dxdu 5
dxdu
5
2251 x
dx
51
12
du
u
215
1
u
duCu 1tan
5
1Cx 5tan
5
1 1
xu dxdu
dxdu
du
udxx
2sec1
duudxx
2sec11
Cux tan
1ln
Cxx
tan
1ln
Even More Difficult Examples
7) Evaluate
8) Evaluate
xdxx cossin 2
dxx
e x
Even More Difficult Examples
7) Evaluate
8) Evaluate
xdxx cossin 2
dxx
e x
xu sin
Even More Difficult Examples
7) Evaluate
8) Evaluate
xdxx cossin 2
dxx
e x
x
duxucos
cos2
xu sindxxdu )(cos
dxx
du
cos
duu2
Cu
3
3
Cx
3
sin3
xu
dxx
du2
1
dxdux )2( duxx
eu)2(
dueu2 Ceu 2 Ce x 2
Difficult Examples (Conclusion)
9) Evaluate
10)Evaluate
dttt 3 54 53
dxxx 12
Difficult Examples (Conclusion)
9) Evaluate
10)Evaluate
dttt 3 54 53
dxxx 12
553 tu dttdu 425
dtt
du
425)
25(
43
14
t
duut duu 3
1
25
1
Cu
3425
1 3
4
Ct 3
45 )53(
100
3
1xudxdu
1ux22 )1( ux1222 uux
duuuu 2
12 )12(
duuuu )2( 2
1
2
3
2
5
Cuuu 2
3
2
5
2
7
3
2
5
4
7
2Cxxx 2
3
2
5
2
7
)1(3
2)1(
5
4)1(
7
2
Homework:
page 371
# 1 – 23 odd
