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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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