X2 T05 05 trig substitutions (2010)

34
Trig Substitutions

Transcript of X2 T05 05 trig substitutions (2010)

Page 1: X2 T05 05 trig substitutions (2010)

Trig Substitutions

Page 2: X2 T05 05 trig substitutions (2010)

Trig Substitutions22 xa tan use ax

Page 3: X2 T05 05 trig substitutions (2010)

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

Page 4: X2 T05 05 trig substitutions (2010)

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

22 ax sec use ax

Page 5: X2 T05 05 trig substitutions (2010)

dx

xai

22

1 e.g.

Page 6: X2 T05 05 trig substitutions (2010)

dx

xai

22

1 e.g.

Page 7: X2 T05 05 trig substitutions (2010)

dx

xai

22

1 e.g.

dadxax

2sectan

222

2

tansec

aada

Page 8: X2 T05 05 trig substitutions (2010)

dx

xai

22

1 e.g.

dadxax

2sectan

222

2

tansec

aada

22

2

secseca

da

dsec

Page 9: X2 T05 05 trig substitutions (2010)

dx

xai

22

1 e.g.

dadxax

2sectan

222

2

tansec

aada

22

2

secseca

da

dsec

c tanseclog

Page 10: X2 T05 05 trig substitutions (2010)

dx

xai

22

1 e.g.

dadxax

2sectan

222

2

tansec

aada

22

2

secseca

da

dsec

c tanseclog

x

a

22 xa

Page 11: X2 T05 05 trig substitutions (2010)

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

Page 12: X2 T05 05 trig substitutions (2010)

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

Page 13: X2 T05 05 trig substitutions (2010)

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

Page 14: X2 T05 05 trig substitutions (2010)

dx

xxii

21

Page 15: X2 T05 05 trig substitutions (2010)

dx

xxii

21

ddx

xcos

sin

Page 16: X2 T05 05 trig substitutions (2010)

dx

xxii

21

ddx

xcos

sin

2sin1cossin d

Page 17: X2 T05 05 trig substitutions (2010)

dx

xxii

21

ddx

xcos

sin

2sin1cossin d

2coscossin d

dsin

Page 18: X2 T05 05 trig substitutions (2010)

dx

xxii

21

ddx

xcos

sin

2sin1cossin d

2coscossin d

dsin

c cos

Page 19: X2 T05 05 trig substitutions (2010)

dx

xxii

21

ddx

xcos

sin

2sin1cossin d

2coscossin d

dsin

c cos

x1

21 x

Page 20: X2 T05 05 trig substitutions (2010)

dx

xxii

21

ddx

xcos

sin

2sin1cossin d

2coscossin d

dsin

c cos

x1

21 x

cx 21

Page 21: X2 T05 05 trig substitutions (2010)

2 3iii x dx

Page 22: X2 T05 05 trig substitutions (2010)

2 3iii x dx2

3 tan

3 sec

x

dx d

Page 23: X2 T05 05 trig substitutions (2010)

2 3iii x dx2

3 tan

3 sec

x

dx d

2

3

3 sec 3 sec

3 sec

d

d

Page 24: X2 T05 05 trig substitutions (2010)

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

Page 25: X2 T05 05 trig substitutions (2010)

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

Page 26: X2 T05 05 trig substitutions (2010)

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

Page 27: X2 T05 05 trig substitutions (2010)

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

Page 28: X2 T05 05 trig substitutions (2010)

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

Page 29: X2 T05 05 trig substitutions (2010)

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

Page 30: X2 T05 05 trig substitutions (2010)

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

Page 31: X2 T05 05 trig substitutions (2010)

22 2 32 3 3 3log

3 3x xx dx x x c

Page 32: X2 T05 05 trig substitutions (2010)

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

Page 33: X2 T05 05 trig substitutions (2010)

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

Page 34: X2 T05 05 trig substitutions (2010)

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