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8.2 Trigonometric Integrals 443 These identities come from the angle sum formulas for the sine and cosine functions (Section 1.3). They give functions whose antiderivatives are easily found. EXAMPLE 8 Evaluate sin 3x cos 5x dx. Solution From Equation (4) with m = 3 and n = 5, we get sin 3x cos 5x dx = ~ j [sin ( — 2x) + sin 8x] dx = I2 (sin 8x + C. Powers of Sines and Cosines Evaluate the integrals in Exercises 1-22. — sin 2x) <ix cos 8x . cos 2x 16 vr + sinxdx cosx 27. / —^^d. V3 VT sin2 20 cos3 20 22. / 3 cos3 20 sin 20 dd cos 2x dx cos x sin x dx sin3 x dx sin5 x dx cos3 x dx 11. 47. 29. / J5TT/ /6 J sin x cos x dx 3 3 IT 1 — sinx 1 — sinx' / cot3 x dx : dx vT JTT/6 /2 flVT- cos 20 dO 31. / J tan x dx49. (1 - 32. / cos203/2 13. ^ cos2 x dx 15. / sin ydy Jo 17. / 8 sin4 x dx Jo , / 16 sin2 x cos2 x dx 8 — 3 sin dx Evaluate the integrals in Exercises 33-50. 4. / sin4 2x cos 2x dx 6. / M/if: Multiply by 37r/4 cos3 4x dx Vl - sin2xdx 30. / JTT/2 sin — 8. / /o dx 2 /-TT/6 3 cos5 3x dx Jo 10. / .J sec2 x tan2 x dx d/9 25. / Vl - sin2 tdt J 34. J sec x tan x dx 36. 11 / e r f 2*am*** 23. / J^^dx 38. 2 y sec3 x CTT/2 16. y sin2 x dx tan3 x dx 38. / sec4 x tan2 7 cos1 tdt 18. / 8 x dx 40. / <?x sec <?x dx 33. / sec2 x tan x dx 35. Evaluate the integrals in Exercises 23-32 42. / 3 sec 3x dx 277 / sec x tan • • 37, 39, 3 cos4 27TX dx sec3 x dx r/3 8 sin >'cos ydy Jo 20. / CTT/2 ! y sec 0d0 4 -./* f ir j2 Jo I; 41. 24. / 26, VT — cos2xdx Jo 44. / sec x dx /71/4 46. / 6 tan4 x dx 7-7T/4 43. / esc4 0 dO 45. / 48. / cot6 2x dx 50. 4 tan3 x <£r / 8cot4/dr