4.5 Integration by Substitution
( ( )) ' )) ( )( (f g x g x d F gx x C Outside Function
Inside FunctionDerivative of
Inside Function
Antidifferentiation of a Composite Function
Official Description
Antidifferentiation of a Composite Function
Let be a continuous function whose range is an interval ,
and let be a function that is continuous on . If is
differentiable on its domain and is an antiderivative of
on , then
g I
f I g
F f
I
( ( )) '( ) ( ( ))f g x g x dx F g x C If ( ), then '( ) and u g x du g x dx
( ) ( ) .f u du F u C Problem
4.5 Integration by Substitution2 2Evaluate ( 1) (2 )x x dx
Composite Function?
Derivative of Inside Function?
3
2 2 2
2
32( 1) (2 )3
1
11
3
2
ux x dx u d x C
u x
du xdx
u C
Substitution
4.5 Integration by Substitution
:Check
22 1 2x x 2 31( 1)
3
dx C
dx
2 2 2 3( ( ))1
( 1) (2 ) ( 1)3
F g xx Cx dx x C
4.5 Integration by Substitution
Evaluate 5cos5xdx5
5
u x
du dx
5cos5 cos sin 5sinxdx udu x Cu C
2
32 2
3232
1
1( 1
2
1)
2 6
1
632
u x
du xdx
u ux x dx u du
xCC C
4.5 Integration by Substitution
2 2Evaluate ( 1)x x dx
4.5 Integration by Substitution
Evaluate 2 1x dx
3/ 2 3/ 3/ 221/ 2
2 1
12 1
2 3/
11
2 32 3
2
2
u x
du dx
u ux dx u du CC
xC
1/ 2
5/ 2 3/ 21/ 2 3/ 2 1/ 2
5/ 2 3/ 2 5/ 2 3 5/ 2 3// 2
Change of Variables or Write in terms o
2 1
?????
1
2 2
1 1 1
2 2 4 4 5 / 2 3/ 2
1 2 2
4 5 3 1
2
2 1 2 1
f
10
06
u x
du dx
u du
u dux dx
u du u uu u u du
x x
x
C
u u u uC C
u
2
6C
4.5 Integration by Substitution
Evaluate 2 1x x dx
4.5 Integration by Substitution2sin 3Evaluate cos3x xdx
??????
sin 3 co 3 3s
u
u x du x dx
2 2
3 3 3
1sin 3 cos3
3
1
3 3
si
99
n 3x
x xdx u du
u uC C C
We do not need change of variables!
4.5 Integration by SubstitutionTHM 4.13 The General Power Rule for Integration
is a differentiable function of g x
1( )
( ) '( ) , 11
nn g x
g x g x dx C nn
Equivalently, ( )u g x
1
, 11
nn uu du C n
n
4.5 Integration by Substitution
Substitution and the General Power Rule4(a) 3(3 1) x dx
5(3 1)
5
xC
2(b) (2 1)( )x x x dx 2 2( )
2
x xC
2 3(c) 3 2x x dx 3 / 232 2
3
xC
2 2
4(d)
(1 2 )
xdx
x
2 1(1 2 )x C
2(e) cos sinx xdx 3cos
3
xC
2cos ( sin )x x dx
22( 2)(f) ?x dx General Power Rule won't work because
the integrand lacks a factor of .xGOTO (f)
4.5 Integration by Substitution
Substitution and the General Power Rule
4 513(a) 1
5 3(3 1)
3 1
3
x d x Cx
u x
du dx
4.5 Integration by Substitution
Substitution and the General Power Rule
2
2
222
(b) (2 1)( )
2 1
1 1
22
x x x dx
u x x
du
x x
x dx
udu C Cu
4.5 Integration by Substitution
Substitution and the General Power Rule
2 3
3
2
1/ 2 3 / 2 3 / 23
(c) 3 2
2
22
3
3
2
3
x x dx
u x
du x dx
u du u C x C
4.5 Integration by Substitution
Substitution and the General Power Rule
2
2 2
2
2 12
1
2
4(d)
(1 2 )
1 2
4
1
1 2 1
xdx
x
u x
du xdx
Cu du u C Cx x
4.5 Integration by Substitution
Substitution and the General Power Rule
3
2
2 3
(e) cos sin
cos
sin
1cos
3
1
3
x xdx
u x
du xdx
u du Cu C x
4.5 Integration by Substitution
2 2( 2)x dx 4 2( 4 4)x x dx 5 34
45 3
x xx C
5 344
5 3
x xx C
So what do we do when substitution
doesn't w
Write the integrand in another for
ork?
m!!!
4.5 Integration by Substitution1
0
2 3Evaluate ( 1)x x dx2 1 2u x du xdx
Now determine the new upper and lower bounds.
Lower Bound Upper Bound0 1x u 1 2x u
1 2 3
0( 1)x x dx
1 2 3
0
1( 1) 2
2x x dx
2 3
1
1
2u du
24
4 4
1
1 12 1
2 4 8
u
15
8
Change of Variables
You can change the bounds and
keep everything in terms of u!
4.5 Integration by Substitution5
1Evaluate
2 1
xA x
xd
2 1 2u x du dx Won't work!
2 1u x 2 2 1u x
dx udu
2 1
2
ux
Differential of x?
Lower Bound Upper Bound1 1x u 5 3x u
5
1 2 -1
xdx
x
23
1
1 1
2
uudu
u
3 2
1
1( 1)
2u du
33
1
1
2 3
uu
1 19 3 1
2 3
16
3
Change of Variables
4.5 Integration by Substitution
THM 4.15 Integration of Even and Odd Functions
- 0
-
Let be integrable on , .
1. is an even function ( ) 2 ( ) .
2. is an odd function ( ) 0
a a
a
a
a
f a a
f f x dx f x dx
f f x dx
/2 3
- /2(sin cos sin cos 0)x x x x dx
4.5 Integration by Substitution/2 3
- /2Evaluate (sin cos sin cos )x x x x dx
3( ) sin ( ) cos( ) sin( ) cos( )f x x x x x
Note: cos( ) cos ,x x 3 3sin ( ) sin ,x x sin( ) sinx x
3( ) sin cos sin cosf x x x x x ( )f x
Even or Odd??
4.5 Integration by Substitution
4.5 Integration by Substitution
HW 4.5/2-6,7-29odd,33-43odd,49,51,59,61,65,67,71,79,80,90,91-94
Welcome to the first abstract part of integration. In today's assignment there
will be times when is appropriate and times when y
r
ou simply need
to or use a ewrite the integrand trigonome
substitution
tric identity
"differential equations
You will also be asked to solve (another branch of mathematics
that engineers study.) That is: giv
(they never
en , solve for
go
.
.
"
away )
dyy
dx
recipes critical
thinking ski
If you are dependent upon , you will struggle. If you have developed the
that are needed to achieve success at the college level, you will struggle on
a few problems
lls
, but still be well on your way to receiving a on the "5" AP Exam.
Who are you?