12 x1 t06 01 integration using substitution (2013)

Integration Using Substitution

Integration Using Substitution

Try to convert the integral into a “standard integral”

Integration Using Substitution

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

dxxn 0 if ,0;1 ,1

1 1

nxnxn

n

Integration Using Substitution

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

dxxn 0 if ,0;1 ,1

1 1

nxnxn

n

dxx1

0 ,ln xx

Integration Using Substitution

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

dxxn 0 if ,0;1 ,1

1 1

nxnxn

n

dxx1

0 ,ln xx

Integration Using Substitution

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

dxxn 0 if ,0;1 ,1

1 1

nxnxn

n

dxx1

0 ,ln xx

dxeax 0 ,1 ae

aax

Integration Using Substitution

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

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

Integration Using Substitution

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

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

dxax sec2 0 ,tan1 aax

a

dxaxax tansec 0 ,sec1 aax

a

dxax sec2 0 ,tan1 aax

a

dxaxax tansec 0 ,sec1 aax

a

dxxa

122 0 ,tan1 1 a

ax

a

dxax sec2 0 ,tan1 aax

a

dxaxax tansec 0 ,sec1 aax

a

dxxa

122 0 ,tan1 1 a

ax

a

dxxa

122 axaa

ax

0, ,sin 1

dxax sec2 0 ,tan1 aax

a

dxaxax tansec 0 ,sec1 aax

a

dxxa

122 0 ,tan1 1 a

ax

a

dxxa

122 axaa

ax

0, ,sin 1

dxax

122 0 ,ln 22 axaxx

dxax sec2 0 ,tan1 aax

a

dxaxax tansec 0 ,sec1 aax

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

“NON” STANDARD INTEGRALSNot listed in the standard integrals, however will save time if you know

“NON” STANDARD INTEGRALSNot listed in the standard integrals, however will save time if you know

duu 2 u

“NON” STANDARD INTEGRALSNot listed in the standard integrals, however will save time if you know

duu 2 u

2

duu

1u

“NON” STANDARD INTEGRALSNot listed in the standard integrals, however will save time if you know

ln xdx ln x x x

duu 2 u

2

duu

1u

“NON” STANDARD INTEGRALSNot listed in the standard integrals, however will save time if you know

ln xdx ln x x x

duu 2 u

2

duu

1u

tan xdx logsec x

“NON” STANDARD INTEGRALSNot listed in the standard integrals, however will save time if you know

ln xdx ln x x x

duu 2 u

2

duu

1u

complementary trig ratio complement of the answer

tan xdx logsec x

“NON” STANDARD INTEGRALSNot listed in the standard integrals, however will save time if you know

ln xdx ln x x x

duu 2 u

2

duu

1u

complementary trig ratio complement of the answer

e.g.cot x dx

tan xdx logsec x

“NON” STANDARD INTEGRALSNot listed in the standard integrals, however will save time if you know

ln xdx ln x x x

duu 2 u

2

duu

1u

complementary trig ratio complement of the answer

e.g.cot x dx

tan xdx logsec x

cot is the complement of tan

“NON” STANDARD INTEGRALSNot listed in the standard integrals, however will save time if you know

ln xdx ln x x x

duu 2 u

2

duu

1u

complementary trig ratio complement of the answer

e.g.cot x dx log cosec x

tan xdx logsec x

cot is the complement of tan

so the answer is minus and the

complement of sec is cosec

“NON” STANDARD INTEGRALSNot listed in the standard integrals, however will save time if you know

ln xdx ln x x x

duu 2 u

2

duu

1u

complementary trig ratio complement of the answer

e.g.cot x dx log cosec x

tan xdx logsec x

logsin xcot is the complement of tan

so the answer is minus and the

complement of sec is cosec

e.g. (i) dxxx 42

e.g. (i) dxxx 4242 xu

e.g. (i) dxxx 4242 xu

u

e.g. (i) dxxx 4242 xu

xdxdu 2u

e.g. (i) dxxx 4242 xu

xdxdu 2udu

21

e.g. (i) dxxx 4242 xu

duu

dxxx

21

2

21

4221

xdxdu 2udu

21

e.g. (i) dxxx 4242 xu

duu

dxxx

21

2

21

4221

xdxdu 2udu

21

cu 23

32

21

e.g. (i) dxxx 4242 xu

duu

dxxx

21

2

21

4221

xdxdu 2udu

21

cu 23

32

21

cu 23

31

e.g. (i) dxxx 4242 xu

duu

dxxx

21

2

21

4221

xdxdu 2udu

21

cu 23

32

21

cu 23

31

cx 23

2 431

e.g. (i) dxxx 4242 xu

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

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