05 substiution rule

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Substitution Rule

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Advance Mathematics about Substitution Rule A Guide for Beginners

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Page 1: 05 Substiution Rule

Substitution Rule

Page 2: 05 Substiution Rule

duufdxxuxgf

havewetervalinthatonThenxurangeoncontinuousisfIf

tervalinsomeonderivativecontinuoushavexuuLet

)()()((

:,)(

)(

Page 3: 05 Substiution Rule

The Differential

dxxfdyxfyLet

)(aldifferentiThe)(:

Page 4: 05 Substiution Rule

Examples

42)(5.

5sin2)(4.5tan2)(3.

52)(2.

52)(1.

:cases following theofeach for aldifferenti theFind

3

4

xxu

xxuxxuxxu

xxu

du

Page 5: 05 Substiution Rule

Solution

3 232

31

3

2

21

21

34

3

2324242)(5.

cos25sin2)(4.sec25tan2)(3.

5252)(2.

852)(1.

:cases following theofeach for aldifferenti theFind

x

dxdxxduxxxu

dxxduxxudxxduxxu

xdxdxxduxxxu

dxxduxxu

du

Page 6: 05 Substiution Rule

Substitution RuleBasic Problems

Page 7: 05 Substiution Rule

Example (1)

3 74

3

)52(

:

x

dxxI

Evaluate

Page 8: 05 Substiution Rule

duu

uxdux

x

dxxI

xdudxdxxdu

xuLet

x

dxxI

37

3 7

33

3 74

3

33

4

3 74

3

81

8

)52(

88

52:

)52(

Page 9: 05 Substiution Rule

cx

cu

cu

duu

3 44

3 4

34

37

)52(

1323

1323

)34(8

1

81

Page 10: 05 Substiution Rule

Example (2)

xdxx

I

Evaluate

7)52(1:

Page 11: 05 Substiution Rule

cxcu

duu

duxux

dxxx

I

duxx

dudxdxxdxxdu

xxuLet

dxxx

I

8)52(

8

1

)52(1

212

5252:

)52(1

88

7

7

7

21

21

21

21

7

Page 12: 05 Substiution Rule

Example (3)

3 7

2

)5tan2(

sec

:

x

dxxI

Evaluate

Page 13: 05 Substiution Rule

duu

u

xdux

x

dxxI

xdudxdxxdu

xuLetx

dxxI

37

3 7

22

3 7

2

22

3 7

2

21

sec2sec

)5tan2(

sec

sec2sec2

5tan2:)5tan2(

sec

Page 14: 05 Substiution Rule

cx

cu

cu

duu

3 4

3 4

34

37

)5tan2(

183

183

)34(2

1

21

Page 15: 05 Substiution Rule

Example (4)

3 7)5sin2(

cos:

x

dxxI

Evaluate

Page 16: 05 Substiution Rule

duu

ux

dux

xdxxI

xdudxdxxdu

xuLetx

dxxI

37

3 7

3 7

3 7

21

cos2cos

)5sin2(cos

cos2cos2

5sin2:)5sin2(

cos

Page 17: 05 Substiution Rule

cx

cu

cu

duu

3 4

3 4

34

37

)5sin2(

183

183

342

1

21

Page 18: 05 Substiution Rule

Example (5)

dxxxI

Evaluate

)42cos(

:87

Page 19: 05 Substiution Rule

cx

cu

duu

xduux

dxxxI

xdudxdxxdu

xuLet

dxxxI

)42sin(161

sin161

cos161

16cos

)42cos(

1616

42:

)42cos(

8

77

87

77

8

87

Page 20: 05 Substiution Rule

Example (6)

dxxxI

Evaluate

)42(sec

:827

Page 21: 05 Substiution Rule

cx

cu

duu

xduux

dxxxI

xdudxdxxdu

xuLet

dxxxI

)42tan(161

tan161

sec161

16sec

)42(sec

1616

42:

)42(sec

8

2

727

827

77

8

827

Page 22: 05 Substiution Rule

Example (7)

dxxxxI

Evaluate

)42tan()42sec(

:887

Page 23: 05 Substiution Rule

cx

cu

duuu

xduuux

dxxxxI

xdudxdxxdu

xuLet

dxxxxI

)42sec(161

sec161

tansec161

16tansec

)42tan()42sec(

1616

42:

)42tan()42sec(

8

77

887

77

8

887

Page 24: 05 Substiution Rule

Example (8)

3 2

33 )42tan()42sec(

:

x

dxxxI

Evaluate

Page 25: 05 Substiution Rule

cx

cu

duuux

duxuu

x

dxxxI

duxx

dudxdxxdu

xxuLet

x

dxxxI

)42sec(23

sec23

tansec232

3tansec

)42tan()42sec(

23

323

12

4242:

)42tan()42sec(

3

3 2

3 2

3 2

33

3 2

32

32

31

3

3 2

33

Page 26: 05 Substiution Rule

Example (9)

dxxxxI

Evaluate

)4sin2tan()4sin2sec(cos

:

Page 27: 05 Substiution Rule

cx

cu

duuu

xduuux

dxxxxI

xdudxdxxdu

xuLet

dxxxxI

)4cos2sec(21

sec21

tansec21

cos2tanseccos

)4sin2tan()4sin2sec(cos

cos2cos2

4sin2:

)4sin2tan()4sin2sec(cos

Page 28: 05 Substiution Rule

Substitution RuleDefinite Integral Case

Page 29: 05 Substiution Rule

Example (1)

1

02)12( x

dxI

Evaluate

Page 30: 05 Substiution Rule

3

12

3

12

1

02

1

02

212

)12(

31211100

22

12:)12(

udu

u

duxdxI

uxux

dudxdxdu

xuLetxdxI

Page 31: 05 Substiution Rule

31)

32(

21

)131(

21

121

121

21

21

3

1

3

1

1

3

1

2

3

12

uu

duu

udu

Page 32: 05 Substiution Rule

Example (2)

1

034

3

)12( xdxxI

Evaluate

Page 33: 05 Substiution Rule

3

13

3

13

33

1

034

3

33

4

1

034

3

818

)12(

31211100

88

12:

)12(

udu

uxdux

xdxxI

uxux

xdudxdxxdu

xuLet

xdxxI

Page 34: 05 Substiution Rule

181)

98(

161

)191(

161

1161

281

81

81

3

12

3

1

2

3

1

3

3

13

uu

duu

udu

Page 35: 05 Substiution Rule

Example (3)

dxxxI

Evaluate

cossin2

0

Page 36: 05 Substiution Rule

duu

xduxuI

ux

uxx

dudxdxxdu

xuLet

dxxxI

1

0

21

1

0

21

2

0

coscos

1)2

sin(2

00sin0cos

cos

sin:

cossin

Page 37: 05 Substiution Rule

32)01(

32

32

23

1

0

31

0

23

1

0

21

uu

duu

Page 38: 05 Substiution Rule

Substitution Rule

More Challenging Problems

Page 39: 05 Substiution Rule

Example (1)

25

:

xdxxI

Evaluate

Page 40: 05 Substiution Rule

Method 1

Page 41: 05 Substiution Rule

duuu

duu

uu

duuxdxxI

uxand

dudxdxdu

xuLetxdxxI

)2(251

225155

225

52

55

25:25

21

21

21

Page 42: 05 Substiution Rule

cxx

cuu

duuu

21

23

21

23

21

21

)25(254)25(

752

)

212

23(

251

)2(251

Page 43: 05 Substiution Rule

25

:

xdxxI

EvaluatetoMethodAnother

Page 44: 05 Substiution Rule

duu

u

duuuxdxx

I

uxand

duudxdxduu

xu

xuLetxdxxI

)2(252

52

52

25

52

52

52

25

25:25

2

2

2

2

Page 45: 05 Substiution Rule

cxx

cxx

cuu

212

3

212

3

3

)25(254

75)25(2

)25(23

)25(252

)23

(252

Page 46: 05 Substiution Rule

Note that the first method can be used to find the integral of any function of the form:f(x) = x(2n-1) (axn+b)k

for any positive integer n and any real number k (where k is not -1) as the following examples show:

Page 47: 05 Substiution Rule

Example (2)

dxxxI

x

dxxI

x

dxxI

dxxxI

dxxxI

EvaluateExamples

1235

3 22

3

3 2

12

12

)42()5(

)42()4(

)42()3(

)42()2(

)42()1(

::

Page 48: 05 Substiution Rule

In all of the first three examples, we let:u = 2x+ 4and so:du = 2dx → dx = du/2andx = (u - 4)/2

Page 49: 05 Substiution Rule

cxx

cuuduuu

duuuduuuI

uxand

dudxdxduxulet

Solution

xdxxI

dxxxI

dxxxI

IntegralsthreefirstThe

13)42(4

14)42(

41

)13

414

(41)4(

41

)4(41

2242

42

242

:

)42()3(

)42()2(

)42()1(

:

1314

13141213

12121

33

122

121

Page 50: 05 Substiution Rule

cxx

cuuduuu

duuuduuuI

uxand

dudxdxduxulet

Solution

xdxxI

dxxxI

dxxxI

)11()42(4

)10()42(

41

)11(4

)10(41)4(

41

)4(41

2242

42

242

:

)42()3(

)42()2(

)42()1(

1110

11101211

12122

33

122

121

Page 51: 05 Substiution Rule

cxx

cuuduuu

duuuduuuI

uxand

dudxdxduxulet

Solution

xdxxI

dxxxI

dxxxI

31

34

31

34

32

31

32

32

3

33

122

121

)42(12)42(43

41

314

344

1)4(41

)4(41

2242

42

242

:

)42()3(

)42()2(

)42()1(

Page 52: 05 Substiution Rule

In the fourth example, we let:u = 2x2+ 4and so:du = 4xdx → dx = du/4xandx2 = (u - 4)/2

Page 53: 05 Substiution Rule

beforeascontinueweThen

duuu

duuu

duuxu

xdux

I

uxand

xdudxdxxdu

xuLet

x

dxxI

)4(81

24

41

414

24

44

42:

)42()4(

32

32

32

2

32

3

4

2

2

3 22

3

4

Page 54: 05 Substiution Rule

In the fifth example, we let:u = 2x3+ 4and so:du = 6x2dx → dx = du/6x2

andx3 = (u - 4)/2

Page 55: 05 Substiution Rule

beforeascontinuethenand

duuu

duuu

duux

xduuxI

uxand

xdudxdxxdu

xuLet

dxxxI

)4(121

61

24

61

6

24

66

42:

)42()5(

12

12

123

2125

5

3

22

3

12355

Page 56: 05 Substiution Rule

Examples (3)

2

0

2

2

2

cos.

sin.

cos.

:

dxxIc

dxxIb

dxxIa

Evaluate

Page 57: 05 Substiution Rule

The double angle formulas can simplify these problems, by replacing cos2x by (1+cos2x)/2 and sin2x by (1- cos2x)/2

dxxIb

dxxIa

onesequivalentfollowingthetotransferedareproblemstheSo

22cos1.

22cos1.

:

Page 58: 05 Substiution Rule

cxx

Whycxx

dxxdx

dxx

dxxIa

2sin41

21

?22sin

21

21

2cos21

21

22cos1

cos. 2

Page 59: 05 Substiution Rule

cxx

Whycxx

dxxdx

dxx

dxxIa

2sin41

21

?22sin

21

21

2cos21

21

22cos1

sin. 2

Page 60: 05 Substiution Rule

4

0sin41)0(

21sin

41

221

2sin41

21

cos.

20

2

0

2

xx

dxxIc

Page 61: 05 Substiution Rule

Note: If the problems were what we have below, then his would be like the basic examples. Do them!

dxxxIb

dxxxIa

Evaluate

cossin.

sincos.

:

2

2

Page 62: 05 Substiution Rule

Other Problems having the same idea: Do them!

xxxxxHdxxx

If

dxxx

Ie

xxxxHdxxxId

xxxxxHdxxxIc

dxxx

Ib

dxxxIa

Evaluate

cos)sin1(coscoscos:int(cossin

3.

sincos

3.

cot)csccsccotcsc:int(cotcsc.

)tansecsectansec:int(tansec.

csccot

3.

sectan.

:

2233

5 2

5 2

21

23

3

99100100

2

5 2

2100