X2 T04 05 trig substitutions (2011)

Trig Substitutions

Trig Substitutions22 xa tan use ax

Trig Substitutions22 xa tan use ax 22 xa cosor sin use axax

Trig Substitutions22 xa tan use ax 22 xa cosor sin use axax

22 ax sec use ax

dx

xai

22

1 e.g.

dx

xai

22

1 e.g.

dx

xai

22

1 e.g.

dadxax

2sectan

222

2

tansec

aada

dx

xai

22

1 e.g.

dadxax

2sectan

222

2

tansec

aada

22

2

secseca

da

dsec

dx

xai

22

1 e.g.

dadxax

2sectan

222

2

tansec

aada

22

2

secseca

da

dsec

c tanseclog

dx

xai

22

1 e.g.

dadxax

2sectan

222

2

tansec

aada

22

2

secseca

da

dsec

c tanseclog

x

a

22 xa

dx

xai

22

1 e.g.

dadxax

2sectan

222

2

tansec

aada

22

2

secseca

da

dsec

c tanseclog

x

a

22 xa

cax

axa

22

log

dx

xai

22

1 e.g.

dadxax

2sectan

222

2

tansec

aada

22

2

secseca

da

dsec

c tanseclog

x

a

22 xa

cax

axa

22

log

caxxa loglog 22

dx

xai

22

1 e.g.

dadxax

2sectan

222

2

tansec

aada

22

2

secseca

da

dsec

c tanseclog

x

a

22 xa

cax

axa

22

log

caxxa loglog 22

cxxa 22log

dx

xxii

21

dx

xxii

21

ddx

xcos

sin

dx

xxii

21

ddx

xcos

sin

2sin1cossin d

dx

xxii

21

ddx

xcos

sin

2sin1cossin d

2coscossin d

dsin

dx

xxii

21

ddx

xcos

sin

2sin1cossin d

2coscossin d

dsin

c cos

dx

xxii

21

ddx

xcos

sin

2sin1cossin d

2coscossin d

dsin

c cos

x1

21 x

dx

xxii

21

ddx

xcos

sin

2sin1cossin d

2coscossin d

dsin

c cos

x1

21 x

cx 21

2 3iii x dx

2 3iii x dx2

3 tan

3 sec

x

dx d

2 3iii x dx2

3 tan

3 sec

x

dx d

2

3

3 sec 3 sec

3 sec

d

d

2 3iii x dx2

3 tan

3 sec

x

dx d

2

3

3 sec 3 sec

3 sec

d

d

23 sec sec d

2 3iii x dx2

3 tan

3 sec

x

dx d

2

3

3 sec 3 sec

3 sec

d

d

23 sec sec d secu

sec tandu d 2secdv d

tanv

2 3iii x dx2

3 tan

3 sec

x

dx d

2

3

3 sec 3 sec

3 sec

d

d

23 sec sec d secu

sec tandu d 2secdv d

tanv

23sec tan 3 sec tan d

2 3iii x dx2

3 tan

3 sec

x

dx d

2

3

3 sec 3 sec

3 sec

d

d

23 sec sec d secu

sec tandu d 2secdv d

tanv

23sec tan 3 sec tan d 33sec tan 3 sec 3 secd

2 3iii x dx2

3 tan

3 sec

x

dx d

2

3

3 sec 3 sec

3 sec

d

d

23 sec sec d secu

sec tandu d 2secdv d

tanv

23sec tan 3 sec tan d 33sec tan 3 sec 3 secd

33sec tan 3 sec 3log sec tand

2 3iii x dx2

3 tan

3 sec

x

dx d

2

3

3 sec 3 sec

3 sec

d

d

23 sec sec d secu

sec tandu d 2secdv d

tanv

23sec tan 3 sec tan d 33sec tan 3 sec 3 secd

33sec tan 3 sec 3log sec tand

x2 3x

3

2 3iii x dx2

3 tan

3 sec

x

dx d

2

3

3 sec 3 sec

3 sec

d

d

23 sec sec d secu

sec tandu d 2secdv d

tanv

23sec tan 3 sec tan d 33sec tan 3 sec 3 secd

33sec tan 3 sec 3log sec tand

x2 3x

32 2

23 33 3 3log3 3 3 3

x x x xx dx

22 2 32 3 3 3log

3 3x xx dx x x c

22 2 32 3 3 3log

3 3x xx dx x x c

2 22 3 3 33 log

2 2 3 3x x x xx dx c

22 2 32 3 3 3log

3 3x xx dx x x c

2 22 3 3 33 log

2 2 3 3x x x xx dx c

2

2 23 33 log 32 2

x xx dx x x c

22 2 32 3 3 3log

3 3x xx dx x x c

2 22 3 3 33 log

2 2 3 3x x x xx dx c

2

2 23 33 log 32 2

x xx dx x x c

Exercise 2E; 1, 2, 3, 5, 6, 7, 9, 13, 17, 19, 20