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4029 u-du: Integrating Composite Functions
AP Calculus
(5 𝑥5+4 𝑥3+3𝑥+2 )5
Find the derivative
5 (5 𝑥5+4 𝑥3+3 𝑥+2 )4 (25 𝑥4+12 𝑥2+3 )
dx/du-part of the antiderivative
u-du SubstitutionIntegrating Composite Functions
(Chain Rule)Revisit the Chain Rule
If let u = inside function
du = derivative of the inside
becomes
2 3( 1)d
xdx
2 3 2 2( 1) 3( 1) (2 )d
x x xdx
3 2( ) = 3( )d duu u dxdx
¿ 𝑥2+1
2 𝑥𝑑𝑥
A Visual Aid
USING u-du Substitution a Visual AidREM: u = inside function du = derivative of the inside
let u =
becomes now only working with f , the outside function
2 23( 1) 2x xdx 23u du
𝑥2+12 𝑥𝑑𝑥
(𝑥2+1 )𝑑𝑢=¿ 2 𝑥𝑑𝑥
3 (𝑢3
3 )+𝑐𝑢3+𝑐
(𝑥2+1 )3+𝑐
Example 1 : du given
Ex 1:2 3(5 1) *10x xdx 𝑢=5 𝑥2+1
𝑑𝑢=10𝑥𝑑𝑥𝑢3𝑑𝑢𝑢4
4+𝑐
14
(5 𝑥2+1 )4+𝑐 proof
14
(5 𝑥2+1 )4+𝑐
4( 14 ) (5 𝑥2+1 )3 (10 𝑥 )
(5 𝑥2+1 )3 (10 𝑥 )
Example 2: du given
Ex 2:
1 22 33 ( 1)x x dx 𝑢=𝑥3+1𝑑𝑢=3 𝑥2𝑑𝑥
(𝑥3+1 )12 3 𝑥2𝑑𝑥
𝑢12 𝑑𝑢
𝑢32
32
+𝑐
23𝑢
32+𝑐
𝑦=23
(𝑥3+1 )32 +𝑐
Example 3: du given
Ex 3:
2
2*
1
xdx
x 𝑢=(𝑥2+1 )
𝑑𝑢=2𝑥𝑑𝑥
(𝑥2+1 )− 1
2 2 𝑥𝑑𝑥
𝑢− 1
2 𝑑𝑢𝑢
12
12
+𝑐
2𝑢12 +𝑐
2 (𝑥2+1 )12 +𝑐
Example 4: du given
Ex 4:
2( ) sec ( )tan x x dx
Both ways !
𝑢=tan(𝑥¿)¿𝑑𝑢=𝑠𝑒𝑐2(𝑥)𝑑𝑥
𝑢𝑑𝑢𝑢2
2+𝑐
12𝑡𝑎𝑛2(𝑥)+𝑐
𝑢=sec (𝑥)𝑑𝑢=sec (𝑥 ) tan (𝑥 )
𝑢𝑑𝑢𝑢2
2+𝑐
12𝑠𝑒𝑐2 (𝑥 )+𝑐
1+𝑡𝑎𝑛2 (𝑥 )=𝑠𝑒𝑐2(𝑥)
Derivative only
𝑠𝑒𝑐2 (𝑥 )𝑑𝑥
Function and derivative
tan (𝑥 ) 𝑠𝑒𝑐2 (𝑥 )𝑑𝑥
Example 5: Regular Method
Ex 5:
2
cos
sin
xdx
x cos (𝑥) (sin (𝑥 ))−2𝑑𝑥
(sin (𝑥 ))−2 cos (𝑥 )𝑑𝑥𝑢−2𝑑𝑢𝑢−1
−1+𝑐 −𝑢−1+𝑐
−sin (𝑥 )−1+𝑐
−1
sin (𝑥 )+𝑐=−csc (𝑥 )+𝑐
𝑢=sin(𝑥 )𝑑𝑢=cos (𝑥 )𝑑𝑥
cos (𝑥)sin (𝑥)
∗ 1sin (𝑥 )
= cot (𝑥 ) csc (𝑥 )𝑑𝑥
− csc (𝑥 )+𝑐
Working with Constants < multiplying by one>
Constant Property of Integration
ILL. let u =
du = and
becomes =
Or alternately = =
5cos 5 cosx dx x dx
4(1 2 )x dx (1 2 )x
4 1( )
2u du
41( )
2u du
2dx
42(1 2 )
2x dx 41
( )2
u du
1
2du dx
41(1 2 ) 2
2x dx
Example 6 : Introduce a Constant - my method
2* 9x x dx
𝑢=9 −𝑥2
𝑑𝑢=−2 𝑥𝑑𝑥
−2−2𝑥 √9− 𝑥2𝑑𝑥
−12−2
−12 (9−𝑥2 )
12 − 2𝑥𝑑𝑥
−12𝑢
12 𝑑𝑢
−12 (𝑢
32
32
)+𝑐−
13𝑢
32 +𝑐
−13
(9−𝑥2 )32 +𝑐
Example 7 : Introduce a Constant
2sec (3 )x dx𝑢=3 𝑥𝑑𝑢=3𝑑𝑥
33 𝑠𝑒𝑐2 (3 𝑥 ) 𝑑𝑥
13 𝑠𝑒𝑐2 (3 𝑥 ) 3𝑑𝑥13 𝑠𝑒𝑐2 (𝑢)𝑑𝑢13
tan (𝑢)+𝑐
13
tan (3 𝑥 )+𝑐
sec (𝑥 ) tan (𝑥 )𝑑𝑥 𝑢=sec 𝑥𝑑𝑢=sec 𝑥 tan𝑥
sec 𝑥sec 𝑥 sec (𝑥 ) tan (𝑥 )𝑑𝑥
1sec 𝑥 sec 𝑥 tan𝑥 sec 𝑥 𝑑𝑥
1sec 𝑥𝑢𝑑𝑢
1sec 𝑥
𝑢2
2+𝑐
( 1sec𝑥 ) 𝑠𝑒𝑐
2𝑥2
+𝑐
12
sec𝑥+𝑐
Example 8 : Introduce a Constant << triple chain>>
4sin (2 )cos(2 )x x dx 𝑢=sin(2𝑥)𝑑𝑢=cos (2 𝑥 )2𝑑𝑥
12 𝑠𝑖𝑛4 (2𝑥 ) cos (2𝑥 )∗2𝑑𝑥12𝑢4𝑑𝑢12 (𝑢
5
5 )+𝑐𝑢5
10+𝑐
110
𝑠𝑖𝑛5 (2𝑥 )+𝑐
Example 9 : Introduce a Constant - extra constant
<< extra constant>
You is what You is inside5 (3 𝑥+4 )5𝑑𝑥
𝑢=3 𝑥+4𝑑𝑢=3𝑑𝑥
13
5 (3 𝑥+4 )5 3𝑑𝑥53𝑢5𝑑𝑢
53 (𝑢
6
6 )+𝑐5
18(3 𝑥+4 )6+𝑐
Example 10: Polynomial
2 4
3 1
(3 2 1)
xdx
x x
𝑢=(3 𝑥2− 2𝑥+1 )𝑑𝑢=(6 𝑥−2)𝑑𝑥
12 2(3 𝑥−1)
(3 𝑥2− 2𝑥+1 )4𝑑𝑥
12𝑢− 4𝑑𝑢
12 (𝑢
−3
− 3 )+𝑐−
16
(3 𝑥2− 2𝑥+1 )− 3+𝑐
Example 11: Separate the numerator
2
2 1
1
xdx
x
𝑢=𝑥2+1𝑑𝑢=2𝑥𝑑𝑥
2 𝑥𝑥2+1
𝑑𝑥+ 1
𝑥2+1
𝑑𝑢𝑢
+ 1
𝑥+12
ln|𝑢|+arctan(𝑥)+𝑐
𝑢−1𝑑𝑢+ 1
𝑥2+1
¿ 𝑢0
0
Formal Change of Variables << the Extra “x”>>
ILL: Let
Solve for x in terms of u then
and becomes
2 6 *2x x dx (2 6)u x
6
2
ux
2du dx6
* *2
uu du
¿ 1
2 (𝑢
32 − 6𝑢
12 )𝑑𝑢
12𝑢3 /2𝑑𝑢−
126𝑢1/2𝑑𝑢= 1
2 (𝑢5/2
52 )
❑
−( 12 )6(𝑢
3 /2
32 )= 1
5𝑢5 /2 −2𝑢3 /2
15
(2 𝑥+6 )52 −2 (2𝑥−6 )3/2+𝑐
Formal Change of Variables << the Extra “x”>>
Rewrite in terms of u - du
2 1
3
xdx
x
(2𝑢−7 )𝑢− 1
2 𝑑𝑢
𝑢=𝑥+3𝑑𝑢=𝑑𝑥
𝑥=𝑢−3
2 𝑥=2𝑢− 6
2 𝑥−1=2𝑢− 7
(2𝑢−32 −7𝑢
−12)𝑑𝑢
2∗ 25𝑢
52 −7∗2𝑢
12 +𝑐
45𝑢
52 −14𝑢
12+𝑐
45
(𝑥+3 )52 −14 (𝑥+3 )
12 +𝑐
Assignment
Day 1 Worksheet Larson HW 4029
Day 2 Basic Integration Rules Wksht
extra x Larson 4029 58f
anti for tan /cot Text p. 338 # 18 - 52 (3x)
Integrating Composite Functions(Chain Rule)
( 1)( ) = n( ) *n ndu u u
dx
Remember: Derivatives Rules
Remember: Layman’s Description of Antiderivatives
( 1)( ) n nn u du u c
*2nd meaning of “du” du is the derivative of an implicit “u”
Development
must have the derivative of the inside in order to find
the antiderivative of the outside
*2nd meaning of “dx” dx is the derivative of an implicit “x” more later if x = f then dx = f /
( ( )) '( ( ))* '( )d
f g x f g x g xdx
( ( )) '( ( ))* '( )d
f g x f g x g xdx
( ( )) [ '( ( ))* '( )]d f g x f g x g x dx
( ( )) '( ( ))* '( )f g x f g x g x dx
Development
from the layman’s idea of antiderivative
“The Family of functions that has the given derivative”
must have the derivative of the inside in order to find
---------- the antiderivative of the outside
( ( )) '( ( ))* '( )d
f g x f g x g xdx
( ( )) '( ( ))* '( )d
f g x f g x g xdx
( ( )) '( ( ))* '( )f g x f g x g x dx
3( )d
udx
23( ) * u du
Working With Constants: Constant Property of Integration
With u-du Substitution
REM: u = inside function du = derivative of the inside
Missing Constant?
2 2 2 23( 1) *2 = 3 ( 1) *2x xdx x xdx 23 u du
Worksheet - Part 1
5cos 5 cosx dx x dx
4(1 2 )x dx u = du =
4 4 42 1 1(1 2 ) = (1 2 ) 2 = ( )
2 2 2x dx x dx u du