12x1 t06 01 integration using substitution (2010)

Integration Using Substitution


Try to convert the integral into a “standard integral”


Try to convert the integral into a “standard integral”STANDARD INTEGRALS

dxxn 0 if ,0;1 ,1

1 1

nxnxn

n



dxxn 0 if ,0;1 ,1

1 1

nxnxn

n

dxx1

0 ,ln xx



dxxn 0 if ,0;1 ,1

1 1

nxnxn

n

dxx1

0 ,ln xx

dxeax 0 ,1 ae

aax



dxxn 0 if ,0;1 ,1

1 1

nxnxn

n

dxx1

0 ,ln xx

dxeax 0 ,1 ae

aax

dxax cos 0 ,sin1 aax

a



dxxn 0 if ,0;1 ,1

1 1

nxnxn

n

dxx1

0 ,ln xx

dxeax 0 ,1 ae

aax

dxax cos 0 ,sin1 aax

a

dxax sin 0 ,cos1 aax

a

dxax sec2 0 ,tan1 aax

a


a

dxaxax tansec 0 ,sec1 aax

a


a


a

dxxa

122 0 ,tan1 1 a

ax

a


a


a

dxxa

122 0 ,tan1 1 a

ax

a

dxxa

122 axaa

ax

0, ,sin 1


a


a

dxxa

122 0 ,tan1 1 a

ax

a

dxxa

122 axaa

ax

0, ,sin 1

dxax

122 0 ,ln 22 axaxx


a


a

dxxa

122 0 ,tan1 1 a

ax

a

dxxa

122 axaa

ax

0, ,sin 1

dxax

122 0 ,ln 22 axaxx

dxax

122 22ln axx

e.g. (i) dxxx 42

e.g. (i) dxxx 4242 xu


u


xdxdu 2u


xdxdu 2udu

21


duu

dxxx

21

2

21

4221

xdxdu 2udu

21


duu

dxxx

21

2

21

4221

xdxdu 2udu

21

cu 23

32

21


duu

dxxx

21

2

21

4221

xdxdu 2udu

21

cu 23

32

21

cu 23

31


duu

dxxx

21

2

21

4221

xdxdu 2udu

21

cu 23

32

21

cu 23

31

cx 23

2 431


duu

dxxx

21

2

21

4221

xdxdu 2udu

21

cu 23

32

21

cu 23

31

cx 23

2 431

cxx 4431 22

dx

xxxii

7841

2

dx

xxxii

7841

2

dxxduxxu

88784 2

dx

xxxii

7841

2

dxxduxxu

88784 2

udu

81

dx

xxxii

7841

2

dxxduxxu

88784 2

cu log81

udu

81

dx

xxxii

7841

2

dxxduxxu

88784 2

cu log81

cxx 784log81 2

udu

81

dx

xxxii

7841

2

dxxduxxu

88784 2

cu log81

cxx 784log81 2

udu

81

3log xx

dxiii

dx

xxxii

7841

2

dxxduxxu

88784 2

cu log81

cxx 784log81 2

udu

81

3log xx

dxiii

xdxdu

xu

log

dx

xxxii

7841

2

dxxduxxu

88784 2

cu log81

cxx 784log81 2

udu

81

3log xx

dxiii

xdxdu

xu

log

duuudu

3

3

dx

xxxii

7841

2

dxxduxxu

88784 2

cu log81

cxx 784log81 2

udu

81

3log xx

dxiii

xdxdu

xu

log

duuudu

3

3

cu

cu

2

2

21

21

dx

xxxii

7841

2

dxxduxxu

88784 2

cu log81

cxx 784log81 2

udu

81

3log xx

dxiii

xdxdu

xu

log

duuudu

3

3

cu

cu

2

2

21

21

c

x 2log2

1

dx

xxiv

1

dx

xxiv

1ududx

xuux2

11 2

dx

xxiv

1ududx

xuux2

11 2

udu

uu 21 2

dx

xxiv

1ududx

xuux2

11 2

duu 12 2

uduuu 21 2

dx

xxiv

1ududx

xuux2

11 2

duu 12 2

cuu

3

312

uduuu 21 2

dx

xxiv

1ududx

xuux2

11 2

duu 12 2

cuu

3

312

uduuu 21 2

cxx 12132 3

dx

xxiv

1ududx

xuux2

11 2

duu 12 2

cuu

3

312

uduuu 21 2

cxx 12132 3

cxxx 121132

dx

xxiv

1ududx

xuux2

11 2

duu 12 2

cuu

3

312

uduuu 21 2

2

3

5 cossin

xdxxv

cxx 12132 3

cxxx 121132

dx

xxiv

1ududx

xuux2

11 2

duu 12 2

cuu

3

312

uduuu 21 2

2

3

5 cossin

xdxxvxdxduxu

cossin

cxx 12132 3

cxxx 121132

dx

xxiv

1ududx

xuux2

11 2

duu 12 2

cuu

3

312

uduuu 21 2

2

3

5 cossin

xdxxvxdxduxu

cossin

cxx 12132 3

cxxx 121132

23

3sin,

3

u

ux

dx

xxiv

1ududx

xuux2

11 2

duu 12 2

cuu

3

312

uduuu 21 2

2

3

5 cossin

xdxxvxdxduxu

cossin

cxx 12132 3

cxxx 121132

23

3sin,

3

u

ux

1 2

sin,2

u

ux

dx

xxiv

1ududx

xuux2

11 2

duu 12 2

cuu

3

312

uduuu 21 2

2

3

5 cossin

xdxxvxdxduxu

cossin

1

23

5duu

cxx 12132 3

cxxx 121132

23

3sin,

3

u

ux

1 2

sin,2

u

ux

dx

xxiv

1ududx

xuux2

11 2

duu 12 2

cuu

3

312

uduuu 21 2

2

3

5 cossin

xdxxvxdxduxu

cossin

1

23

5duu

123

6

61 u

cxx 12132 3

cxxx 121132

23

3sin,

3

u

ux

1 2

sin,2

u

ux

dx

xxiv

1ududx

xuux2

11 2

duu 12 2

cuu

3

312

uduuu 21 2

2

3

5 cossin

xdxxvxdxduxu

cossin

1

23

5duu

123

6

61 u

38437

231

61

66

cxx 12132 3

cxxx 121132

23

3sin,

3

u

ux

1 2

sin,2

u

ux

4

3225 x

xdxvi

4

3225 x

xdxvixdxdu

xu2

25 2

4

3225 x

xdxvixdxdu

xu2

25 2

9,416,3

uxux

4

3225 x

xdxvixdxdu

xu2

25 2

16

9

21

9

16

21

21

duu

udu

9,416,3

uxux

4

3225 x

xdxvixdxdu

xu2

25 2

16

9

21

9

16

21

21

duu

udu

16

9

21

u

9,416,3

uxux

4

3225 x

xdxvixdxdu

xu2

25 2

16

9

21

9

16

21

21

duu

udu

16

9

21

u

1916

9,416,3

uxux

5

0

225 dxxvii

5

0

225 dxxvii

ududx

xuux

cos55

sinsin5 1

5

0

225 dxxvii

ududx

xuux

cos55

sinsin5 1

2

1sin,50

0sin,0

1

1

u

uxuux

5

0

225 dxxvii

ududx

xuux

cos55

sinsin5 1

2

0

2 cos5sin2525

uduu

2

1sin,50

0sin,0

1

1

u

uxuux

5

0

225 dxxvii

ududx

xuux

cos55

sinsin5 1

2

0

2 cos5sin2525

uduu

2

0

2

2

0

2

cos25

cos5cos25

udu

uduu

2

1sin,50

0sin,0

1

1

u

uxuux

5

0

225 dxxvii

ududx

xuux

cos55

sinsin5 1

2

0

2 cos5sin2525

uduu

2

0

2

2

0

2

cos25

cos5cos25

udu

uduu

2

0

2

0

2sin21

225

2cos1225

uu

duu2

1sin,50

0sin,0

1

1

u

uxuux

5

0

225 dxxvii

ududx

xuux

cos55

sinsin5 1

2

0

2 cos5sin2525

uduu

2

0

2

2

0

2

cos25

cos5cos25

udu

uduu

2

0

2

0

2sin21

225

2cos1225

uu

duu2

1sin,50

0sin,0

1

1

u

uxuux

425

0sin21

2225

5

0

225 dxxvii

5

0

225 dxxviiy

x5

5

-5

225 xy

5

0

225 dxxviiy

x5

5

-5

225 xy

425

541 2

Exercise 6C; 1, 2ace, 3, 4ace, 5ace etc, 6, 7 & 8ac, 11, 12Exercise 6D; 1, 2a, 3, 4b, 5ac, 6ace etc, 7ab(i,ii)

8ab (i,iii,vi), 9ab (ii,iv,vi), 11, 13

Patel Exercise 2A ace in all NOTE: substitution is not given in Extension 2

5

0

225 dxxviiy

x5

5

-5

225 xy

425

541 2

12x1 t06 01 integration using substitution (2010)

Education

x aa1 x

dx ln x x

duu sin x x

x cos xdxu sin x x

x1 x u

u log x xlog x

u262 x

u c3 x