Trigonometric IntegralsRecall Basic IdentitiesIntegral of sinn x, n OddSlide 4Integral of sinn x, n EvenCombinations of sin, cosSlide 7Combinations of tanm, secnSlide 9Integrals of Even Powers of sec, cscWallis's FormulasSlide 12AssignmentTrigonometric IntegralsLesson 8.3Recall Basic Identities•Pythagorean Identities•Half-Angle Formulas2 22 22 2sin cos 1tan 1 sec1 cot cscq qq qq q+ =+ =+ =221 cos 2sin21 cos 2cos2qqqq-=+=These will be used to integrate powers of sin and cosThese will be used to integrate powers of sin and cosIntegral of sinn x, n Odd•Split into product of an even and sin x•Make the even power a power of sin2 x•Use the Pythagorean identity•Let u = cos x, du = -sin x dx5 4sin sin sinx dx x x dx= �� �( )24 2sin sin sin sinx x dx x x dx� =� �( ) ( )2 22 2sin sin 1 cos sinx x dx x x dx= -� �( )22 2 41 1 2 ...u du u u du- - =- - + =� �Integral of sinn x, n Odd•Integrate and un-substitute•Similar strategy with cosn x, n odd2 4 3 53 52 11 23 52 1cos cos cos3 5u u du u u u Cx x C- - + =- + - +=- + - +�Integral of sinn x, n Even•Use half-angle formulas•Try Change to power of cos2 x•Expand the binomial, then integrate21 cos 2sin2qq-=4cos 5x dx�( )( )2221cos 5 1 cos102x dx x dx� �= +� �� �� �Combinations of sin, cos•General form•If either n or m is odd, use techniques as beforeSplit the odd power into an even power and power of oneUse Pythagorean identitySpecify u and du, substituteUsually reduces to a polynomialIntegrate, un-substitute sin cosm nx x dx��Try withTry with2 3sin cosx x dx��Combinations of sin, cos•Consider•Use Pythagorean identity•Separate and use sinn x strategy for n odd3 2sin 4 cos 4x x dx��( ) ( )3 2 3 5sin 4 1 sin 4 sin 4 sin 4x x dx x x dx� - = -� �Combinations of tanm, secn•When n is evenFactor out sec2 xRewrite remainder of integrand in terms of Pythagorean identity sec2 x = 1 + tan2 xThen u = tan x, du = sec2x dx•Try4 3sec tany y dyCombinations of tanm, secn•When m is oddFactor out tan x sec x (for the du)Use identity sec2 x – 1 = tan2 x for even powers of tan xLet u = sec x, du = sec x tan x•Try the same integral with this strategy4 3sec tany y dyNote similar strategies for integrals involving combinations ofcotm x and cscn xNote similar strategies for integrals involving combinations ofcotm x and cscn xIntegrals of Even Powers of sec, csc•Use the identity sec2 x – 1 = tan2 x •Try 4sec 3x dx 2 22 22 2 23sec 3 sec 31 tan 3 sec 3sec 3 tan 3 sec 31 1tan 3 tan 33 9x x dxx x dxx x x dxx x C Wallis's Formulas•If n is odd and (n ≥ 3) then•If n is even and (n ≥ 2) then/ 20/ 202 4 6 1cos3 5 71 3 5 1cos2 4 6 2nnnx dxnnx dxnppp-= �������-= ������� ���And … Believe it or notThese formulas are also valid if cosnx is replaced by sinnxAnd … Believe it or notThese formulas are also valid if cosnx is replaced by sinnxWallis's Formulas•Try it out … / 250cos x dxp�/ 270sin x dxp�Assignment •Lesson 8.3•Page 540•Exercises 1 – 41
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