# 08 - CHAPTER 8 Integration Techniques LHpitals Rule and...

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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

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95 CHAPTER 8 Integration Techniques, L’Hôpital’s Rule, and Improper Integrals Section 8.1 Basic Integration Rules 2. (a) (b) (c) (d) matches (a). E x x 2 1 1 dx d dx f ln s x 2 1 1 d 1 C g 5 2 x x 2 1 1 d dx f arctan x 1 C g 5 1 1 1 x 2 5 2 s 1 2 3 x 2 d s x 2 1 1 d 3 d dx 3 2 x s x 2 1 1 d 2 1 C 4 5 s x 2 1 1 d 2 s 2 d 2 s 2 x ds 2 ds x 2 1 1 ds 2 x d s x 2 1 1 d 4 d dx f ln ! x 2 1 1 1 C g 5 1 2 1 2 x x 2 1 1 ± 5 x x 2 1 1 1. (a) (b) (c) (d) matches (b). E x ! x 2 1 1 dx d dx f ln s x 2 1 1 d 1 C g 5 2 x x 2 1 1 5 x 2 ! x 2 1 1 d dx 3 1 2 ! x 2 1 1 1 C 4 5 1 2 1 1 2 ± s x 2 1 1 d 2 1 y 2 s 2 x d d dx f ! x 2 1 1 1 C g 5 1 2 s x 2 1 1 d 2 1 y 2 s 2 x d 5 x ! x 2 1 1 5 2 x ! x 2 1 1 d dx f 2 ! x 2 1 1 1 C g 5 2 1 1 2 ± s x 2 1 1 d 2 1 y 2 s 2 x d 3. (a) (b) (c) (d) matches (c). E 1 x 2 1 1 dx d dx f ln s x 2 1 1 d 1 C g 5 2 x x 2 1 1 d dx f arctan x 1 C g 5 1 1 1 x 2 d dx 3 2 x s x 2 1 1 d 2 1 C 4 5 s x 2 1 1 d 2 s 2 d 2 s 2 x ds 2 ds x 2 1 1 ds 2 x d s x 2 1 1 d 4 5 2 s 1 2 3 x 2 d s x 2 1 1 d 3 d dx f ln ! x 2 1 1 1 C g 5 1 2 1 2 x x 2 1 1 ± 5 x x 2 1 1 4. (a) (b) (c) (d) matches (c). E x cos s x 2 1 1 d dx d dx f 2 2 x sin s x 2 1 1 d 1 C g 52 2 x f cos s x 2 1 1 ds 2 x dg 2 2 sin s x 2 1 1 d 2 f 2 x 2 cos s x 2 1 1 d 1 sin s x 2 1 1 dg d dx 3 1 2 sin s x 2 1 1 d 1 C 4 5 1 2 cos s x 2 1 1 ds 2 x d 5 x cos s x 2 1 1 d d dx 3 2 1 2 sin s x 2 1 1 d 1 C 4 1 2 cos s x 2 1 1 ds 2 x d x cos s x 2 1 1 d d dx f 2 x sin s x 2 1 1 d 1 C dg 5 2 x f cos s x 2 1 1 ds 2 x dg 1 2 sin s x 2 1 1 d 5 2 f 2 x 2 cos s x 2 1 1 d 1 sin s x 2 1 1 dg
96 Chapter 8 Integration Techniques, L’Hôpital’s Rule, and Improper Integrals 5. Use E u n du . u 5 3 x 2 2, du 5 3 dx , n 5 4 E s 3 x 2 2 d 4 dx 6. Use E du u . u 5 t 2 2 t 1 2, du 5 s 2 t 2 1 d dt E 2 t 2 1 t 2 2 t 1 2 dt 7. Use E du u . u 5 1 2 2 ! x , du 52 1 ! x dx E 1 ! x s 1 2 2 ! x d dx 8. Use E du u 2 1 a 2 . u 5 2 t 2 1, du 5 2 dt , a 5 2 E 2 s 2 t 2 1 d 2 1 4 dt 9. Use E du ! a 2 2 u 2 . u 5 t , du 5 dt , a 5 1 E 3 ! 1 2 t 2 dt 10. Use E u n du . u 5 x 2 2 4, du 5 2 x dx , n 1 2 E 2 2 x ! x 2 2 4 dx 11. Use E sin u du . u 5 t 2 , du 5 2 t dt E t sin t 2 dt 12. Use E sec u tan u du . u 5 3 x , du 5 3 dx E sec 3 x tan 3 x dx 13. Use E e u du . u 5 sin x , du 5 cos x dx E s cos x d e sin x dx 14. Use E du u ! u 2 2 a 2 . u 5 x , du 5 dx , a 5 2 E 1 x ! x 2 2 4 dx 15. Let 5 s x 2 4 d 6 1 C E 6 s x 2 4 d 5 dx 5 6 E s x 2 4 d 5 dx 5 6 s x 2 4 d 6 6 1 C u 5 x 2 4, du 5 dx .

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