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, cscAssignment ANew Patterns for the IntegrandExampleFinishing UpKnowing Which SubstitutionTry It!!Keep Going!ApplicationAssignment BTrigonometric IntegralsLesson 7.3Recall 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 beforeSplit the odd power into an even power and power of oneUse Pythagorean identitySpecify u and du, substituteUsually reduces to a polynomialIntegrate, 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 odd( ) ( )3 2 3 5sin 4 1 sin 4 sin 4 sin 4x x dx x x dx� - = -� �3 2sin 4 cos 4x x dx��Combinations of tanm, secn•When n is evenFactor out sec2 xRewrite remainder of integrand in terms of Pythagorean identity sec2 x = 1 + tan2 xThen u = tan x, du = sec2x dx•Try4 3sec tany y dyCombinations of tanm, secn•When m is oddFactor out tan x sec x (for the du)Use identity sec2 x – 1 = tan2 x for even powers of tan xLet u = sec x, du = sec x tan x•Try the same integral with this strategy4 3sec tany y dyNote 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     Assignment A•Lesson 7.3A•Page 447•Exercises 5 – 31 oddNew Patterns for the Integrand•Now we will look for a different set of patterns•And we will use them in the context of a right triangle•Draw and label the other two triangles which show the relationships of a and x2 2 2 2 2 2a x a x x a- + -ax2 2a x+Example•Given•Consider the labeled triangleLet x = 3 tan θ (Why?)And dx = 3 sec2 θ dθ•Then we have29dxx +�3x2 23 x+θ223sec9 tan 9dq qq +�Use identitytan2x + 1 = sec2xUse identitytan2x + 1 = sec2x23sec3 sec ln sec tan3secdd Cq qq q q qq= = = + +� �Finishing Up•Our results are in terms of θWe must un-substitute back into xUse the triangle relationshipsln sec tan Cq q+ +3x2 23 x+θ29ln3 3x xC++ +Knowing Which Substitutionuu2 2u a-Try It!!•For each problem, identify which substitution and which triangle should be used3 29x x dx-�221 xdxx-�22 5x x dx- +�( )24 1x dx- -�Keep Going!•Now finish the integration3 29x x dx-�221 xdxx-�22 5x x dx- +�( )24 1x dx- -�Application•Find the arc length of the portion of the parabola y = 10x – x2 that is above the x-axis•Recall the arc length formula[ ]21 '( )biaL f x dx= +�Assignment B•Lesson 7.3B•Page 447•Exercises 33 – 53