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8.2 Trigonometric Integrals Page 1/5 THOMAS’ CALCULUS (12/E) 8.2 Trigonometric Integrals 開課班級: (105-2) 通訊 1/電機 1/智財學程 微積分 授課教師: 吳漢銘 (國立臺北大學統計學系 副教授) 教學網站: http://www.hmwu.idv.tw 學號: 系級: 1 1.1 姓名: Products of Powers of Sines and Cosines sinm x cosn x dx odd ⇒m= sinm x = sin2k+1 x = even and n is odd cosn x = cos2k+1 x Case 1. If m is Case 2. If m is Case 3. If both m and n are 2k + 1 = even , use sin2 x = 1 − cos2 x (sin2 x)k sin x . ⇒n= 2k + 1 (cos2 x)k cos x ⇒ use sin2 x = = = (1 − cos2 x)k sin x , use cos2 x = 1 − sin2 x (1 − sin2 x)k cos x 1 − cos 2x 2 , cos2 x = 1 + cos 2x 2 Ex. 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (example1, p444) Evaluate sin3 x cos2 x dx. sol: (105-2) 通訊 1/電機 1/智財學程 微積分 January 29, 2017 . 8.2 Trigonometric Integrals Page 2/5 Ex. 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (example2, p445) Evaluate cos5 x dx. sol: Ex. 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (example3, p445) Evaluate sin2 x cos4 x dx. sol: 2 Eliminating Square Roots 2.1 Use cos2 θ = (1 + cos 2θ)/2 Ex. 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (example4, p446) π/4 Evaluate √ 1 + cos 4x dx. 0 sol: (105-2) 通訊 1/電機 1/智財學程 微積分 January 29, 2017 8.2 Trigonometric Integrals 3 Page 3/5 Integrals of Powers of tan x and sec x 3.1 Use tan2 x = sec2 x − 1 3.2 Use sec2 x = tan2 x + 1 Ex. 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (example5, p446) Evaluate tan4 x dx. sol: Ex. 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (example6, p447) Evaluate sec3 x dx. sol: (105-2) 通訊 1/電機 1/智財學程 微積分 January 29, 2017 8.2 Trigonometric Integrals 4 Page 4/5 Products of Sine and Cosines 4.1 sin mx sin nx = 4.2 sin mx cos nx = 4.3 cos mx cos nx = 1 [cos(m − n)x − cos(m + n)x] . 2 1 [sin(m − n)x + sin(m + n)x] . 2 1 [cos(m − n)x + cos(m + n)x] . 2 Ex. 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (example7, p448) Evaluate sin 3x cos 5x dx. sol: (105-2) 通訊 1/電機 1/智財學程 微積分 January 29, 2017 8.2 Trigonometric Integrals Page 5/5 實習課練習 (EXERCISE 8.2) π sin5 8. 0 x dx 2 7 cos7 t dt 16. π 8 sin4 x dx 17. 0 π/2 sin2 2θ cos3 2θ dθ 22. 0 2π r 23. 0 25. πp 1 − cos x dx 2 1 − sin2 t dt 0 π/2 31. √ θ 1 − cos 2θ dθ 0 sec3 x tan x dx 35. sec4 x tan2 x dx 38. π 53. sin 3x sin 3x dx 55. −π cos 3x cos 4x dx 57. sin2 θ cos 3θ dθ (105-2) 通訊 1/電機 1/智財學程 微積分 January 29, 2017