8.4 Trigonometric Integrals Powers of Sine and Cosine

sin cosn mu udu∫sin cos cos sinn nu udu u udu∫ ∫

2 2sin 1 cosu u= −1. If n is odd, leave one sin u factor and use

for all other factors of sin.

2 2cos 1 sinu u= −2. If m is odd, leave one cos u factor and use

for all other factors of cos.

2 21sin (1 cos 1cos (1 cos2 ) 2 )22

oru u uu = += −

3. If both powers is even, use power reducing formulas:

and take the substitution v = cos u.

and make the substitution v = sin u.

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Powers of sin and cos

3sin (2 )dθ θ∫

2 3sin ( ) cos ( )dθ θ θ∫

2 2sin ( ) cos ( )dθ θ θ∫

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Powers of sin and cos3 2 2sin (2 ) sin 2 sin 2 (1 cos 2 )sin 2d d dθ θ θ θ θ θ θ θ= = −∫ ∫ ∫

2 3(sin 2 cos 2 sin 2 1 1cos 2 cos2 6

) 2 Cdθ θ θ θ θ θ− +− = +∫

=∫ θθθ d32 cossin =∫ θθθθ dcoscossin 22

( ) =−∫ θθθθ dcossin1sin 22

( ) ( ) =−∫ θθθ sin sin1sin 22 d

( ) ( ) =−∫ θθθ sin sinsin 42 d C+− θθ 53 sin51sin

31

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Powers of sin and cos

2 2 1 1sin ( )cos ( ) (1 cos 2 ) (1 cos 2 )2 2

d dθ θ θ θ θ θ= − +∫ ∫

21 1 1(1 cos 2 ) (1 (1 cos 4 )4 4 21 1 1 1(1 cos 4 ) ( 4 )4 2 4 2

d d

d cos d

θ θ θ θ

θ θ θ θ

− = − +

− − = −

∫ ∫

∫ ∫=θθ−−∫ d)4cos21

211(

41

=θθ−∫ d)4cos21

21(

41

=θθ−∫ d)4cos1(81 C+θ−

θ 4sin321

8

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Example :

4sin cos x x dx⋅∫

( )4sin cos x x dx∫ Let sinu x=

cos du x dx=4 u du∫51

5u C+

51 sin5

x C+

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3 2cos sinx xdx∫22 si os n sco cx x xdx∫

( )2 21 sin sin cosx x xdx−∫2 4sin cos sin cosx xdx x xdx−∫

3 5cos cos3 5

x x C− +

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Example :3sin

cos x dxx

⋅∫Because you expect to use the power rule with the cos term, save one of the sin(x) for the du part and use trig to convert the rest

1/ 2 2(cos ) sin (sin )x x x dx−∫1/ 2 2(cos ) sin (1 cos )x x x dx− −∫

( )1/ 2 3 / 2(cos ) sin (cos )sinx x x x dx− −∫Ziad Zahreddine

( )1/ 2 3 / 2(cos ) sin (cos )sinx x x x dx− −∫1/ 2 3 / 2( 1) ( 1) ((cos ) sin (cos1) ( 1 in)) sx xdx x xdx−− − −∫−−∫

5 / 21/ 2 2(cos )2(cos )

5xx C− + +

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Example : 4sin xdx∫Powers are even – use Identities

2 1 cos2sin2

xx −=

21 cos22

x dx−⎛ ⎞⎜ ⎟⎝ ⎠∫

21 (1 2cos2 cos 2 )4

x x dx− +∫Do it again for the last term

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4sin xdx∫21 (1 2cos2 )2

4cosx xx d− +∫

11 (1 2cos2 s4 )4

co2

x dx x−− +∫

1 1 12cos2 cos44 2 2

dx xdx dx xdx− ∫ + ∫ − ∫∫ 21

8 8+

sin 2 1 sin 44 4 8 32x x x x C− + − ++

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Tangents and secants2tan sec sec sec tann nu udu u u udu∫ ∫

sec tann mu udu∫These identities are useful.

2 2 2 2tan sec 1 tan 1 secorθ θ θ θ= − + =

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sec tann mu udu∫1. If m is odd, leave one sec u tan u factor and use

for all other factors of tan.and take the substitution v = sec u.

1sectan 22 −= xx

2. If n is even, leave one sec2 u factor and usefor all other factors of sec.

and make the substitution v = tan u.1tansec 22 += xx

3. If m even and n odd, usefor all other factors of tan to get

integrals in odd powers of sec,1sectan 22 −= xx

then integrate by parts.Ziad Zahreddine

∫ xdxx 33 tansecExample

∫ dxxxxx )tan(sectansec 22

Take u = sec x and use tan2 x = sec2 x - 1

du = sec x tan x dx

=−∫ dxxxxx )tan)(sec1(secsec 22 ∫ − duuu )1( 22

∫ −= duuu )( 24 Cuu+−=

35

35

Cxx+−=

3sec

5sec 35

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∫ xdxx 44 tansecExample

∫ dxxxx )(sectansec 242

Take u = tan x and use sec2 x = tan2 x + 1

du = sec2 x dx

∫ + duuu 42 )1(

∫ += duuu )( 46 Cuu++=

57

57

Cxx++=

5tan

7tan 55

=+∫ dxxxx )(sectan)1(tan 242

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∫ xdxx 43 tansecExample

Use tan2 x = sec2 x - 1

=−∫ dxxx 223 )1(secsec ∫ +− dxxxx )1sec2(secsec 243

∫ +−= dxxxx )secsec2(sec 357

Integration by parts may be used to integrate odd powers of sec x.

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∫ xdx3secExample

u = sec x, dv = sec2 xWe integration by parts

du = sec x tan x dx, v = tan x

=∫ xdx3sec ∫− )tan)(sec(tantansec xdxxxxx

∫−= xdxxxx sectantansec 2

∫ −−= xdxxxx sec)1(sectansec 2

∫ −−= dxxxxx )sec(sectansec 3

∫∫ −+= xdxxdxxx 3secsectansec

∫∫ += xdxxxxdx sectansecsec2 3

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∫ xdx3secExample

∫∫ += xdxxxxdx sectansecsec2 3

∫∫ += xdxxxxdx sec21tansec

21sec3

Cxxxxxdx +++=∫ |tansec|ln21tansec

21sec3

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∫ xdx4tanExample

=∫ xdx4tan =⋅∫ xdxx 22 tantan ∫ −⋅ dxxx )1(sectan 22

∫∫ −⋅= xdxxdxx 222 tansectan

∫∫ −−⋅= dxxxdxx )1(secsectan 222

∫ ∫∫ +−⋅= dxxdxxdxx 222 secsectan

Cxxx ++−= tantan31 3

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Products of Sines and Cosines

∫ dxnxmx )sin()sin( ∫ dxnxmx )cos()sin(

∫ dxnxmx )cos()cos(

The following trigonometric identities will be useful.

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Sum and Difference FormulasSum and Difference Formulas

sin( ) sin cos cos sinA B A B A B+ = +

sin( ) sin cos cos sinA B A B A B− = −

cos( ) cos cos sin sinA B A B A B+ = −

cos( ) cos cos sin sinA B A B A B− = +

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Product- identitiesProduct- identities

sin sin Α Α coscos Β Β = = 22sin (sin (Α+Β) + Α+Β) + sin (sin (ΑΑ−−Β) Β)

sin sin Α Α sin sin Β Β = = 22coscos ((ΑΑ−−Β) Β) −− coscos ((ΑΑ++Β) Β)

coscos Α Α coscos Β Β = = 22coscos ((Α+Β) Α+Β) ++ coscos ((ΑΑ−−Β) Β)

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∫ xdxx 5cos3sinExample

∫ −+ dxxx )]2sin()8[sin(21

=∫ xdxx 5cos3sin

∫ −= dxxx )]2sin()8[sin(21

Cxx++−=

42cos

168cos

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Section 8.4, P. 585

Assignment: 1, 5, 11, 22, 25, 31, 33, 37

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