Trig Integrals

Trig Integrals(1) Standard Integrals

caxa

axdx cos1sin

Trig Integrals(1) Standard Integrals

caxa

axdx cos1sin

caxa

axdx sin1cos

Trig Integrals(1) Standard Integrals

caxa

axdx cos1sin

caxa

axdx sin1cos

caxa

axdx tan1sec2

Trig Integrals(1) Standard Integrals

caxa

axdx cos1sin

caxa

axdx sin1cos

caxa

axdx tan1sec2

dxaxaxaxdx

cossintan

Trig Integrals(1) Standard Integrals

caxa

axdx cos1sin

caxa

axdx sin1cos

caxa

axdx tan1sec2

dxaxaxaxdx

cossintan

caxa

ORcaxa

seclog1 coslog1

xx nn cosor sin2

xx nn cosor sin2

xdxsin cx cos

xx nn cosor sin2

xdxsin

xdx2sin

cx cos

xx nn cosor sin2

xdxsin

xdx2sin dxx2cos121

cx cos

xx nn cosor sin2

xdxsin

xdx2sin dxx2cos121

cxx

2sin

21

21

cx cos

xx nn cosor sin2

xdxsin

xdx2sin dxx2cos121

cxx

2sin

21

21

cx cos

xdx3sin

xx nn cosor sin2

xdxsin

xdx2sin dxx2cos121

cxx

2sin

21

21

cx cos

xdx3sin xdxx 2sinsin

xx nn cosor sin2

xdxsin

xdx2sin dxx2cos121

cxx

2sin

21

21

cx cos

xdx3sin xdxx 2sinsin

dxxx 2cos1sin

xx nn cosor sin2

xdxsin

xdx2sin dxx2cos121

cxx

2sin

21

21

cx cos

xdx3sin xdxx 2sinsin

dxxx 2cos1sinxdxdu

xusin

cos

xx nn cosor sin2

xdxsin

xdx2sin dxx2cos121

cxx

2sin

21

21

cx cos

xdx3sin xdxx 2sinsin

dxxx 2cos1sinxdxdu

xusin

cos

duu21

xx nn cosor sin2

xdxsin

xdx2sin dxx2cos121

cxx

2sin

21

21

cx cos

xdx3sin xdxx 2sinsin

dxxx 2cos1sinxdxdu

xusin

cos

duu21

cxx

cuu

coscos3131

3

3

xx nn cosor sin2

xdxsin

xdx2sin dxx2cos121

cxx

2sin

21

21

cx cos

xdx3sin xdxx 2sinsin

dxxx 2cos1sinxdxdu

xusin

cos

duu21

cxx

cuu

coscos3131

3

3

xuxx

xx

PowerOddsome power

cosUsecos1sin Substitute

sinsin as Factorise22

2

xdx4sin

xdx4sin dxx 22sin

xdx4sin dxx 22sin

dxx 22cos141

xdx4sin dxx 22sin

dxx 22cos141

dxxx 2cos2cos2141 2

xdx4sin dxx 22sin

dxx 22cos141

dxxx 2cos2cos2141 2

dxxx 4cos1

212cos21

41

xdx4sin dxx 22sin

dxx 22cos141

dxxx 2cos2cos2141 2

dxxx 4cos1

212cos21

41

dxxx 4cos

212cos2

23

41

xdx4sin dxx 22sin

dxx 22cos141

dxxx 2cos2cos2141 2

dxxx 4cos1

212cos21

41

dxxx 4cos

212cos2

23

41

cxxx

4sin

812sin

23

41

xdx4sin dxx 22sin

dxx 22cos141

dxxx 2cos2cos2141 2

dxxx 4cos1

212cos21

41

dxxx 4cos

212cos2

23

41

cxxx

4sin

812sin

23

41

xx

x

PowerEvenpowersome

2cos121sin Substitute

sin as Factorise

2

2

xdx5sin

xdx5sin dxxx 22sinsin

xdx5sin dxxx 22sinsin

dxxx 22cos1sin

xdx5sin dxxx 22sinsin

dxxx 22cos1sinxdxdu

xusin

cos

xdx5sin dxxx 22sinsin

dxxx 22cos1sinxdxdu

xusin

cos

duuu

duu42

22

21

1

xdx5sin dxxx 22sinsin

dxxx 22cos1sinxdxdu

xusin

cos

duuu

duu42

22

21

1

cxxx

cuuu

53

53

cos51cos

32cos

51

32

xx nn cos and sin3

xx nn cos and sin3

Usually done by substitution xuxu cosor sin

xx nn cos and sin3

Usually done by substitution xuxu cosor sin

xdxxi 35 sincos e.g.

xx nn cos and sin3

Usually done by substitution xuxu cosor sin

xdxxi 35 sincos e.g.

xdxxx sincos1cos 25

xx nn cos and sin3

Usually done by substitution xuxu cosor sin

xdxxi 35 sincos e.g.

xdxduxu

sincos

xdxxx sincos1cos 25

xx nn cos and sin3

Usually done by substitution xuxu cosor sin

xdxxi 35 sincos e.g.

xdxduxu

sincos

xdxxx sincos1cos 25

duuu

duuu57

25 1

xx nn cos and sin3

Usually done by substitution xuxu cosor sin

xdxxi 35 sincos e.g.

xdxduxu

sincos

xdxxx sincos1cos 25

duuu

duuu57

25 1

cxx

cuu

68

68

cos61cos

81

61

81

xx nn cos and sin3

Usually done by substitution xuxu cosor sin

xdxxi 35 sincos e.g.

xdxduxu

sincos

xdxxx sincos1cos 25

duuu

duuu57

25 1

cxx

cuu

68

68

cos61cos

81

61

81

powerhigher eUsually th aseither Choose

u

oddpowersBoth

xdxxii 36 cossin

xdxxii 36 cossin

xdxxx cossin1sin 26

xdxxii 36 cossin

xdxxx cossin1sin 26 xdxduxu

cossin

xdxxii 36 cossin

xdxxx cossin1sin 26 xdxduxu

cossin

duuu

duuu86

26 1

xdxxii 36 cossin

xdxxx cossin1sin 26 xdxduxu

cossin

duuu

duuu86

26 1

cxx

cuu

97

97

sin91sin

71

91

71

xdxxii 36 cossin

xdxxx cossin1sin 26 xdxduxu

cossin

duuu

duuu86

26 1

cxx

cuu

97

97

sin91sin

71

91

71

uevenpoweroneoddpowerOne

aseven Choose&

xdxxii 36 cossin

xdxxx cossin1sin 26 xdxduxu

cossin

duuu

duuu86

26 1

cxx

cuu

97

97

sin91sin

71

91

71

uevenpoweroneoddpowerOne

aseven Choose&

xdxxiii 22 cossin

xdxxii 36 cossin

xdxxx cossin1sin 26 xdxduxu

cossin

duuu

duuu86

26 1

cxx

cuu

97

97

sin91sin

71

91

71

uevenpoweroneoddpowerOne

aseven Choose&

xdxxiii 22 cossin dxxx 22 sin1sin

xdxxii 36 cossin

xdxxx cossin1sin 26 xdxduxu

cossin

duuu

duuu86

26 1

cxx

cuu

97

97

sin91sin

71

91

71

uevenpoweroneoddpowerOne

aseven Choose&

xdxxiii 22 cossin dxxx 22 sin1sin

dxxx 42 sinsin

xdxxii 36 cossin

xdxxx cossin1sin 26 xdxduxu

cossin

duuu

duuu86

26 1

cxx

cuu

97

97

sin91sin

71

91

71

uevenpoweroneoddpowerOne

aseven Choose&

xdxxiii 22 cossin dxxx 22 sin1sin

dxxx 42 sinsin

cxxxxx 4sin3212sin

41

832sin

41

21

xdxxii 36 cossin

xdxxx cossin1sin 26 xdxduxu

cossin

duuu

duuu86

26 1

cxx

cuu

97

97

sin91sin

71

91

71

uevenpoweroneoddpowerOne

aseven Choose&

xdxxiii 22 cossin dxxx 22 sin1sin

dxxx 42 sinsin

cxxxxx 4sin3212sin

41

832sin

41

21

cxx 4sin321

81

xdxxii 36 cossin

xdxxx cossin1sin 26 xdxduxu

cossin

duuu

duuu86

26 1

cxx

cuu

97

97

sin91sin

71

91

71

uevenpoweroneoddpowerOne

aseven Choose&

xdxxiii 22 cossin dxxx 22 sin1sin

dxxx 42 sinsin

cxxxxx 4sin3212sin

41

832sin

41

21

cxx 4sin321

81xx

xxevenpowersBoth

22

22

sin1cosor cos1sin Use

xx nn cotor tan4

xx nn cotor tan4

xdxtan cx coslog

xx nn cotor tan4

xdxtan cx coslog

xdx2tan

xx nn cotor tan4

xdxtan cx coslog

xdx2tan dxx 1sec2

xx nn cotor tan4

xdxtan cx coslog

xdx2tan dxx 1sec2

cxx tan

xx nn cotor tan4

xdxtan cx coslog

xdx2tan dxx 1sec2

cxx tan

xdx3tan

xx nn cotor tan4

xdxtan cx coslog

xdx2tan dxx 1sec2

cxx tan

xdx3tan

xdxxdxx

dxxx

tansectan

1sectan2

2

xx nn cotor tan4

xdxtan cx coslog

xdx2tan dxx 1sec2

cxx tan

xdx3tan

xdxxdxx

dxxx

tansectan

1sectan2

2

xdxduxu

2sectan

xx nn cotor tan4

xdxtan cx coslog

xdx2tan dxx 1sec2

cxx tan

xdx3tan

xdxxdxx

dxxx

tansectan

1sectan2

2

xdxduxu

2sectan

xdxudu tan

xx nn cotor tan4

xdxtan cx coslog

xdx2tan dxx 1sec2

cxx tan

xdx3tan

xdxxdxx

dxxx

tansectan

1sectan2

2

xdxduxu

2sectan

xdxudu tan

cxx

cxu

coslogtan21

coslog21

2

2

xdx4tan

xdx4tan

xdxxdxx

dxxx222

22

tansectan

1sectan

xdx4tan

xdxxdxx

dxxx222

22

tansectan

1sectan

xdxduxu

2sectan

xdx4tan

xdxxdxx

dxxx222

22

tansectan

1sectan

xdxduxu

2sectan

xdxduu 22 tan

xdx4tan

xdxxdxx

dxxx222

22

tansectan

1sectan

xdxduxu

2sectan

xdxduu 22 tan

cxxx

cxxu

tantan31

tan31

3

3

xx nn cosecor sec5

xx nn cosecor sec5

xdxsec

xx nn cosecor sec5

xdxsec

dx

xxxxx

tansectansecsec

xx nn cosecor sec5

xdxsec

dx

xxxxx

tansectansecsec

dxxx

xxxtansec

tansecsec2

xx nn cosecor sec5

xdxsec

dx

xxxxx

tansectansecsec

dxxx

xxxtansec

tansecsec2

cxx tanseclog

xx nn cosecor sec5

xdxsec

dx

xxxxx

tansectansecsec

xdx2sec cx tan

dxxx

xxxtansec

tansecsec2

cxx tanseclog

xx nn cosecor sec5

xdxsec

dx

xxxxx

tansectansecsec

xdx2sec cx tan

xdx3sec

dxxx

xxxtansec

tansecsec2

cxx tanseclog

xx nn cosecor sec5

xdxsec

dx

xxxxx

tansectansecsec

xdx2sec cx tan

xdx3sec xdxx 2secsec

dxxx

xxxtansec

tansecsec2

cxx tanseclog

xx nn cosecor sec5

xdxsec

dx

xxxxx

tansectansecsec

xdx2sec cx tan

xdx3sec xdxx 2secsec xu sec

dxxx

xxxtansec

tansecsec2

cxx tanseclog

xx nn cosecor sec5

xdxsec

dx

xxxxx

tansectansecsec

xdx2sec cx tan

xdx3sec xdxx 2secsec xu sec

dxxx

xxxtansec

tansecsec2

cxx tanseclog

xdxxdu tansec

xx nn cosecor sec5

xdxsec

dx

xxxxx

tansectansecsec

xdx2sec cx tan

xdx3sec xdxx 2secsec xu sec

dxxx

xxxtansec

tansecsec2

cxx tanseclog

xdxxdu tansec xdxdv 2sec

xx nn cosecor sec5

xdxsec

dx

xxxxx

tansectansecsec

xdx2sec cx tan

xdx3sec xdxx 2secsec xu sec

dxxx

xxxtansec

tansecsec2

cxx tanseclog

xdxxdu tansec xdxdv 2sec

xv tan

xx nn cosecor sec5

xdxsec

dx

xxxxx

tansectansecsec

xdx2sec cx tan

xdx3sec xdxx 2secsec xu sec

xdxxxx 2tansectansec

dxxx

xxxtansec

tansecsec2

cxx tanseclog

xdxxdu tansec xdxdv 2sec

xv tan

xx nn cosecor sec5

xdxsec

dx

xxxxx

tansectansecsec

xdx2sec cx tan

xdx3sec xdxx 2secsec xu sec

xdxxxx 2tansectansec

dxxx

xxxtansec

tansecsec2

cxx tanseclog

xdxxdu tansec xdxdv 2sec

xv tan

xxxdxxx

xdxxdxxx

dxxxxx

tanseclogsectansec

secsectansec

1secsectansec

3

3

2

xx nn cosecor sec5

xdxsec

dx

xxxxx

tansectansecsec

xdx2sec cx tan

xdx3sec xdxx 2secsec xu sec

xdxxxx 2tansectansec

cxxxxxdx tanseclogtansecsec2 3

dxxx

xxxtansec

tansecsec2

cxx tanseclog

xdxxdu tansec xdxdv 2sec

xv tan

xxxdxxx

xdxxdxxx

dxxxxx

tanseclogsectansec

secsectansec

1secsectansec

3

3

2

xx nn cosecor sec5

xdxsec

dx

xxxxx

tansectansecsec

xdx2sec cx tan

xdx3sec xdxx 2secsec xu sec

xdxxxx 2tansectansec

cxxxxxdx tanseclogtansecsec2 3

dxxx

xxxtansec

tansecsec2

cxx tanseclog

xdxxdu tansec xdxdv 2sec

xv tan

xxxdxxx

xdxxdxxx

dxxxxx

tanseclogsectansec

secsectansec

1secsectansec

3

3

2

cxxxxxdx tanseclog21tansec

21sec3

xx nn cosecor sec5

xdxsec

dx

xxxxx

tansectansecsec

xdx2sec cx tan

xdx3sec xdxx 2secsec xu sec

xdxxxx 2tansectansec

cxxxxxdx tanseclogtansecsec2 3

dxxx

xxxtansec

tansecsec2

cxx tanseclog

xdxxdu tansec xdxdv 2sec

xv tan

xxxdxxx

xdxxdxxx

dxxxxx

tanseclogsectansec

secsectansec

1secsectansec

3

3

2

cxxxxxdx tanseclog21tansec

21sec3

partsby DonepowersOdd

xdx4sec

xdx4sec dxxx 22 tan1sec

xdx4sec dxxx 22 tan1sec

xdxduxu

2sectan

xdx4sec dxxx 22 tan1sec

xdxduxu

2sectan

duu21

xdx4sec dxxx 22 tan1sec

xdxduxu

2sectan

duu21

cxx

cuu

3

3

tan31tan

31

xdx4sec dxxx 22 tan1sec

xdxduxu

2sectan

duu21

cxx

cuu

3

3

tan31tan

31

xuxx

xx

PowerEvenpowersome

tan Usetan1sec Substitute

secsec as Factorise22

22

xdx4sec dxxx 22 tan1sec

xdxduxu

2sectan

duu21

cxx

cuu

3

3

tan31tan

31

Exercise 2C; 1, 2, 4, 5, 7, 8, 10, 12, 14, 16, 17, 18

xuxx

xx

PowerEvenpowersome

tan Usetan1sec Substitute

secsec as Factorise22

22