CHAPTER 8 Integration Techniques, L’Hôpital’s Rule, and Improper Integrals Section 8.1 Basic Integration Rules . 95 Section 8.2 Integration by Parts . 106 Section 8.3 Trigonometric Integrals . 128 Section 8.4 Trigonometric Substitution . 141 Section 8.5 Partial Fractions . 161 Section 8.6 Integration by Tables and Other Integration Techniques . 173 Section 8.7 Indeterminate Forms and L’Hôpital’s Rule . 184 Section 8.8 Improper Integrals . 199 Review Exercises . 212 Problem Solving . 223 CHAPTER 8 Integration Techniques, L’Hôpital’s Rule, and Improper Integrals Section 8.1 Basic Integration Rules d 1 Ϫ ͞ d 1 2x x 1. (a) ͓2Ίx2 ϩ 1 ϩ C͔ ϭ 2΂ ΃͑x2 ϩ 1͒ 1 2͑2x͒ 2. (a) ͓lnΊx2 ϩ 1 ϩ C͔ ϭ ΂ ΃ ϭ dx 2 dx 2 x2 ϩ 1 x2 ϩ 1 2x d 2x ͑x2 ϩ 1͒2͑2͒ Ϫ ͑2x͒͑2͒͑x2 ϩ 1͒͑2x͒ ϭ (b) ΄ ϩ C΅ ϭ Ίx2 ϩ 1 dx ͑x2 ϩ 1͒2 ͑x2 ϩ 1͒4 2 d 1 Ϫ ͞ x 2͑1 Ϫ 3x ͒ (b) ͓Ίx2 ϩ 1 ϩ C͔ ϭ ͑x2 ϩ 1͒ 1 2͑2x͒ ϭ ϭ dx 2 Ίx2 ϩ 1 ͑x2 ϩ 1͒3 d 1 1 1 Ϫ ͞ d 1 (c) ΄ Ίx2 ϩ 1 ϩ C΅ ϭ ΂ ΃͑x2 ϩ 1͒ 1 2͑2x͒ (c) ͓arctan x ϩ C͔ ϭ dx 2 2 2 dx 1 ϩ x2 x d 2x ϭ (d) ͓ln͑x2 ϩ 1͒ ϩ C͔ ϭ 2Ίx2 ϩ 1 dx x2 ϩ 1 d 2x x (d) ͓ln͑x2 ϩ 1͒ ϩ C͔ ϭ ͵ dx matches (a). dx x2 ϩ 1 x2 ϩ 1 x ͵ dx matches (b). Ίx2 ϩ 1 d 1 2x x 3. (a) ͓lnΊx2 ϩ 1 ϩ C͔ ϭ ΂ ΃ ϭ dx 2 x2 ϩ 1 x2 ϩ 1 d 2x ͑x2 ϩ 1͒2͑2͒ Ϫ ͑2x͒͑2͒͑x2 ϩ 1͒͑2x͒ 2͑1 Ϫ 3x2͒ (b) ΄ ϩ C΅ ϭ ϭ dx ͑x2 ϩ 1͒2 ͑x2 ϩ 1͒4 ͑x2 ϩ 1͒3 d 1 (c) ͓arctan x ϩ C͔ ϭ dx 1 ϩ x2 d 2x (d) ͓ln͑x2 ϩ 1͒ ϩ C͔ ϭ dx x2 ϩ 1 1 ͵ dx matches (c). x2 ϩ 1 d 4. (a) ͓2x sin͑x2 ϩ 1͒ ϩ C͔͒ ϭ 2x͓cos͑x2 ϩ 1͒͑2x͔͒ ϩ 2 sin͑x2 ϩ 1͒ ϭ 2͓2x2 cos͑x2 ϩ 1͒ ϩ sin͑x2 ϩ 1͔͒ dx d 1 1 (b) ΄Ϫ sin͑x2 ϩ 1͒ ϩ C΅ ϭϪ cos͑x2 ϩ 1͒͑2x͒ ϭϪx cos͑x2 ϩ 1͒ dx 2 2 d 1 1 (c) ΄ sin͑x2 ϩ 1͒ ϩ C΅ ϭ cos͑x2 ϩ 1͒͑2x͒ ϭ x cos͑x2 ϩ 1͒ dx 2 2 d (d) ͓Ϫ2x sin͑x2 ϩ 1͒ ϩ C͔ ϭϪ2x͓cos͑x2 ϩ 1͒͑2x͔͒ Ϫ 2 sin͑x2 ϩ 1͒ ϭϪ2͓2x2 cos͑x2 ϩ 1͒ ϩ sin͑x2 ϩ 1͔͒ dx ͵x cos͑x2 ϩ 1͒ dx matches (c). 95 96 Chapter 8 Integration Techniques, L’Hôpital’s Rule, and Improper Integrals 2t Ϫ 1 1 5. ͵͑3x Ϫ 2͒4 dx 6. ͵ dt 7. ͵ dx t2 Ϫ t ϩ 2 Ίx͑1 Ϫ 2Ίx ͒ u ϭ 3x Ϫ 2, du ϭ 3 dx, n ϭ 4 u ϭ t2 Ϫ t ϩ 2, du ϭ ͑2t Ϫ 1͒ dt 1 u ϭ 1 Ϫ 2Ίx, du ϭϪ dx Ί du x Use ͵un du. Use ͵ . u du Use ͵ . u 2 3 Ϫ2x 8. ͵ dt 9. ͵ dt 10. ͵ dx ͑2t Ϫ 1͒2 ϩ 4 Ί1 Ϫ t2 Ίx2 Ϫ 4 u ϭ 2t Ϫ 1, du ϭ 2 dt, a ϭ 2 u ϭ t, du ϭ dt, a ϭ 1 1 u ϭ x2 Ϫ 4, du ϭ 2x dx, n ϭϪ 2 du du Use ͵ . Use ͵ . u2 ϩ a2 Ί 2 Ϫ 2 a u Use ͵un du. 11. ͵t sin t 2 dt 12. ͵sec 3x tan 3x dx 13. ͵͑cos x͒e sin x dx u ϭ t 2, du ϭ 2t dt u ϭ 3x, du ϭ 3 dx u ϭ sin x, du ϭ cos x dx Use ͵sin u du. Use ͵sec u tan u du. Use ͵eu du. 1 14. ͵ dx 15. Let u ϭ x Ϫ 4, du ϭ dx. xΊx2 Ϫ 4 ͑x Ϫ 4͒6 u ϭ x, du ϭ dx, a ϭ 2 ͵6͑x Ϫ 4͒5 dx ϭ 6͵͑x Ϫ 4͒5 dx ϭ 6 ϩ C 6 du Use ͵ . ϭ ͑x Ϫ 4͒6 ϩ C uΊu2 Ϫ a2 16. Let u ϭ t Ϫ 9, du ϭ dt. 17. Let u ϭ z Ϫ 4, du ϭ dz. 2 Ϫ2 5 ͑z Ϫ 4͒Ϫ4 ͵ dt ϭ 2͵͑t Ϫ 9͒Ϫ2 dt ϭ ϩ C ͵ dz ϭ 5͵͑z Ϫ 4͒Ϫ5 dz ϭ 5 ϩ C ͑t Ϫ 9͒2 t Ϫ 9 ͑z Ϫ 4͒5 Ϫ4 Ϫ5 ϭ ϩ C 4͑z Ϫ 4͒4 1 1 18. Let u ϭ t3 Ϫ 1, du ϭ 3t2 dt. 19. ͵ ΄v ϩ ΅ dv ϭ ͵v dv ϩ ͵͑3v Ϫ 1͒Ϫ3͑3͒ dv ͑3v Ϫ 1͒3 3 1 ͵ t 2Ί3 t3 Ϫ 1 dt ϭ ͵͑t 3 Ϫ 1͒1͞3͑3t 2͒ dt 1 1 3 ϭ v2 Ϫ ϩ C 2 6͑3v Ϫ 1͒2 1 ͑t 3 Ϫ 1͒4͞3 ϭ ϩ C 3 4͞3 ͑t 3 Ϫ 1͒4͞3 ϭ ϩ C 4 3 3 20. ͵ ΄x Ϫ ΅ dx ϭ ͵x dx Ϫ ͵ ͑2x ϩ 3͒Ϫ2͑2͒ dx 21. Let u ϭϪt3 ϩ 9t ϩ 1, du ϭ ͑Ϫ3t2 ϩ 9͒ dt ϭ ͑2x ϩ 3͒2 2 Ϫ3͑t2 Ϫ 3͒ dt. x2 3 ͑2x ϩ 3͒Ϫ1 ϭ Ϫ ϩ C t 2 Ϫ 3 1 Ϫ3͑t 2 Ϫ 3͒ 2 2 Ϫ1 ͵ dt ϭϪ ͵ dt Ϫt3 ϩ 9t ϩ 1 3 Ϫt3 ϩ 9t ϩ 1 x2 3 ϭ ϩ ϩ C 1 2 2͑2x ϩ 3͒ ϭϪ lnԽϪt3 ϩ 9t ϩ 1Խ ϩ C 3 Section 8.1 Basic Integration Rules 97 x2 1 22. Let u ϭ x2 ϩ 2x Ϫ 4, du ϭ 2͑x ϩ 1͒ dx. 23. ͵ dx ϭ ͵͑x ϩ 1͒ dx ϩ ͵ dx x Ϫ 1 x Ϫ 1 ϩ x 1 1 Ϫ ͞ ͵ dx ϭ ͵͑x2 ϩ 2x Ϫ 4͒ 1 2͑2͒͑x ϩ 1͒ dx 1 Ίx2 ϩ 2x Ϫ 4 2 ϭ x2 ϩ x ϩ lnԽx Ϫ 1Խ ϩ C 2 ϭ Ίx2 ϩ 2x Ϫ 4 ϩ C 2x 8 24. ͵ dx ϭ ͵2 dx ϩ ͵ dx 25. Let u ϭ 1 ϩ ex, du ϭ ex dx. x Ϫ 4 x Ϫ 4 ex ϭ 2x ϩ 8 lnԽx Ϫ 4Խ ϩ C ͵ dx ϭ ln͑1 ϩ ex͒ ϩ C 1 ϩ ex 1 1 1 1 1 1 26. ͵΂ Ϫ ΃ dx ϭ ͵ ͑3͒ dx Ϫ ͵ ͑3͒ dx 3x Ϫ 1 3x ϩ 1 3 3x Ϫ 1 3 3x ϩ 1 1 1 1 3x Ϫ 1 ϭ lnԽ3x Ϫ 1Խ Ϫ lnԽ3x ϩ 1Խ ϩ C ϭ ln ϩ C 3 3 3 Խ3x ϩ 1Խ 4 4 x 27. ͵͑1 ϩ 2x2͒2 dx ϭ ͵͑4x 4 ϩ 4x2 ϩ 1͒ dx ϭ x5 ϩ x3 ϩ x ϩ C ϭ ͑12x 4 ϩ 20x2 ϩ 15͒ ϩ C 5 3 15 1 3 3 3 1 3 1 1 1 28. ͵x΂1 ϩ ΃ ϭ ͵x΂1 ϩ ϩ ϩ ΃ dx ϭ ͵΂x ϩ 3 ϩ ϩ ΃ dx ϭ x2 ϩ 3x ϩ 3 lnԽxԽ Ϫ ϩ C x x x2 x3 x x2 2 x 1 29. Let u ϭ 2␲x2, du ϭ 4␲x dx. 30. ͵sec 4x dx ϭ ͵sec͑4x͒͑4͒ dx 4 1 ͵x͑cos 2␲ x2͒ dx ϭ ͵͑cos 2␲ x2͒͑4␲x͒ dx 1 4␲ ϭ lnԽsec 4x ϩ tan 4xԽ ϩ C 4 1 ϭ sin 2␲x2 ϩ C 4␲ 31. Let u ϭ ␲x, du ϭ ␲ dx. 32. Let u ϭ cos x, du ϭϪsin x dx. 1 sin x ͵csc͑␲x͒ cot͑␲x͒ dx ϭ ͵csc͑␲x͒ cot͑␲x͒␲ dx ͵ dx ϭϪ͵͑cos x͒Ϫ1͞2͑Ϫsin x͒ dx ␲ Ίcos x ϭϪ Ί ϩ ϭϪ1 ͑␲ ͒ ϩ 2 cos x C ␲ csc x C 33. Let u ϭ 5x, du ϭ 5 dx. 34. Let u ϭ cot x, du ϭϪcsc2 x dx. 1 1 ͵e5x dx ϭ ͵e5x͑5͒ dx ϭ e5x ϩ C ͵csc2 xecot x dx ϭϪ͵ecot x͑Ϫcsc2 x͒ dx ϭϪecot x ϩ C 5 5 5 1 eϪx 35. Let u ϭ 1 ϩ e x, du ϭ e x dx. 36. ͵ dx ϭ 5͵΂ ΃΂ ΃ dx 3ex Ϫ 2 3ex Ϫ 2 eϪx 2 1 e x ͵ dx ϭ 2͵΂ ΃΂ ΃ dx eϪx eϪx ϩ 1 eϪx ϩ 1 e x ϭ 5͵ dx 3 Ϫ 2eϪx e x ϭ 2͵ dx 5 1 1 ϩ e x ϭ ͵ ͑2eϪx͒ dx 2 3 Ϫ 2eϪx ϭ 2 ln͑1 ϩ e x͒ ϩ C 5 ϭ lnԽ3 Ϫ 2eϪxԽ ϩ C 2 98 Chapter 8 Integration Techniques, L’Hôpital’s Rule, and Improper Integrals ln x2 1 ͑ln x͒2 Ϫsin x 37. ͵ dx ϭ 2͵͑ln x͒ dx ϭ 2 ϩ C ϭ ͑ln x͒2 ϩ C 38. Let u ϭ ln͑cos x͒, du ϭ dx ϭϪtan x dx. x x 2 cos x ͵ ͑tan x͒͑ln cos x͒ dx ϭϪ͵ ͑ln cos x͒͑Ϫtan x͒ dx Ϫ͓ln͑cos x͔͒2 ϭ ϩ C 2 1 ϩ sin x 1 ϩ sin x 1 Ϫ sin x 39. ͵ dx ϭ ͵ и dx Alternate Solution: cos x cos x 1 Ϫ sin x 1 ϩ sin x 1 Ϫ sin2 x ͵ dx ϭ ͵͑sec x ϩ tan x͒ dx ϭ ͵ dx cos x cos x͑1 Ϫ sin x͒ ϭ lnԽsec x ϩ tan xԽ ϩ lnԽsec xԽ ϩ C cos2 x ϭ ͵ dx cos x͑1 Ϫ sin x͒ ϭ lnԽsec x͑sec x ϩ tan x͒Խ ϩ C Ϫcos x ϭϪ͵ dx 1 Ϫ sin x ϭϪlnԽ1 Ϫ sin xԽ ϩ C, ͑u ϭ 1 Ϫ sin x͒ 1 ϩ cos ␣ 40.