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# Class Note for MATH 1432 at UH 2

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This 5 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 17 views.

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Date Created: 02/06/15

fsinmz cosnz dz7 n gt 1 odd u sin zSubstitution sinm zoos zdz7 n gt 1 odd 2 2 Set u sinz7 du coszdzi Note cos z 17 sin z sinm z cos z dz sinm z cosn 1 z cos z dz n71 sinm z cos2 z T cos z dz sinm 11 7 sin2 I um1 7 uQVTi1 du polynomial W71 2 ds1nz n71 Write cosnz cosn lzcoszi Write cosn lz cos2 Use 0st 1 7 sin2 z u sin zSubstitution Ezample 1 sin2zcos5zdz u217 113 du u2 7 2114 u52 du 1 3 2 5 1 7 7u 7 7u 7u C 3 5 l 7 l 1 2 1 3 2 4 sinz 777s1nz 7s1nz C 3 5 7 gt f sinm z cos z dz7 m gt 1 odd u cos zSubstitution sinm zoos zdz7 m gt 1 odd Set u cos z7 du 7 sinz dzi Note sin2 z 17 cos2 z sinm z cos z dz sinm 1 z sin z cos z dz L4 sin2 z 2 cosnzsinz dz L4 717 cos2z 2 cosnzdcosz 717 u2mT71un du polynomial 1 1 2 2 Write sinm z sinm z cos zi Write sinm z sin z 2 Use sin z 1 7 cos2 z u cos zSubstitution Ezample 2 sin3zcos2zdz 717 u2u2 du u47u2du 1 1 3115 7 gug C 1 1 Cos3zltgcos2z7 C 12 Even Powers Power Reduction Power Reduction Formulas 1 1 sinzcosz 7 sin2z7 sin2z E 7 Ecos2z7 cos2z E Ecos2z Ezamples 3 1 1 2 7 7 7 COS zdz7lt2 2cos2zgt dz 1 1 7 z1s1n2zC 2 1 1 2 Cos4zdzCos2 zdzlt Cos2zgt dz 1 1 1 2 7lt1 Cos2zlcos 2zgt dz 1 1 1 1 1 71z1s1n2z1lt Cos4zgtdz7 2 sin2z cos2zdz ltsinzcoszgt dz 1 lsin22zdz 1 1 1 Ecos4zgt dz 1 1 z7 sm4zC 13 Powers of Sine or Cosine Reduction by Parts Reduction Formulas 7 n SlnnIdI 77s1n 1zcoser n 7 1 s1n 2 z dz n 7 1 7 cos 2 z dz 1 n cosnzdz 700s 1zs1nz n n Proof Set u sinni1 z d1 sinzdz du n 7 1 sin sin z dz sin 1 sin z7 dz 7 sinn 1 zcosz n 71sinn 2 zcos2 z dz quot 2zcoszdz 39u 7coszv 7 sinn 1 zcosz n 71sinn 2 z17 sin2 z dz 7 sin 1 zcosz n 71sin 2 z17 sin2 z dz 7 n71sinnzdz ngglgnd solve the equation for f sin z dz sin2zdz7lsinzcoszl dz 2 7isinzcosz zC 5 1 4 4 3 s1n zdz7gs1n zcoszg Sln zdz 7 1 4 4 1 2 2 d 7 5s1n zcosz 5 3s1n zcosz 3 s1nz z 77sin4zcosz775in2zcosz7 coszC 15 14 Products of Sine and Cosine Products of Sine and Cosine ProducttoSum Formulas sinAcosB sinA 7 B sinA B sinAsinB cosA 7 B 7 COSA B cosAcosB cosA 7 B CosA B Ezample 5 sin 5z sin 2z dz cos5z 7 2z 7 0055z 2z dz 1 E cos 3z 7 cos 7z dz 1 1 7651n3z7 51n7zc 15 Summary Summary Summary sinm 1 cos 1 dz 0 Odd power of sine substitute u cos I du 7 sinz dz 0 Odd power of cosine substitute u sin I du cosz dz 2 1 0 Both powers even Reduce powemnz cosz E s1n 21 s1n z E 7 1 1 i cos 21 cos2 I E i cos 21 or integration by parts 0 Remember the formulas cos2 I sin2 I 1 sinz y sinzcosy coszsiny cosz y coszcosy 7 sinzsiny Out line Contents 1 Sine and Cosine 1 1 Odd Powers i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 112 Even Powers i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 13 Powers of Sine or Cosine i i i i i i i i i i i i i i i i i i i i i i i 114 Products of Sine and Cosine i i i i i i i i i i i i i i i i i i i i i 115 Summary i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i U1gtJgtCA3CADHH Lecture 8Section 83 Powers and Products of Trigonometric Functions Jiwen He 1 Sine and Cosine 11 Odd Powers fsinmx cosz dz u sin xSubstitution sinmzcoszdz Set u sinz7 du cosz dz 0 For m y 71 sinmzcoszdz umdu um1 C m1 1 m1sinm11C o Form717 sinilzcoszdzu 1duln u C ln sinz C f sin 1 cos 1 dz u cos xSubstitution sin 1 cos 1 dz Setucoszdu7sinzdz o Forn7 717 1 sinzcosnzdziundu7munlc 7n1cosn1zC o Forn717 sinzcosilzdziuildu7ln u C 7ln 00sz C

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