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# ODD04 - CHAPTER Integration Section 4.1 Section 4.2 Section...

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CHAPTER 4 Integration Section 4.1 Antiderivatives and Indefinite Integration . . . . . . . . . 177 Section 4.2 Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Section 4.3 Riemann Sums and Definite Integrals . . . . . . . . . . . 188 Section 4.4 The Fundamental Theorem of Calculus . . . . . . . . . . 192 Section 4.5 Integration by Substitution . . . . . . . . . . . . . . . . . 197 Section 4.6 Numerical Integration . . . . . . . . . . . . . . . . . . . 204 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

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177 CHAPTER 4 Integration Section 4.1 Antiderivatives and Indefinite Integration Solutions to Odd-Numbered Exercises 1. d dx 1 3 x 3 1 C 2 5 d dx s 3 x 2 3 1 C d 52 9 x 2 4 5 2 9 x 4 3. d dx 1 1 3 x 3 2 4 x 1 C 2 5 x 2 2 4 5 s x 2 2 ds x 1 2 d 5. Check: d dt f t 3 1 C g 5 3 t 2 y 5 t 3 1 C dy dt 5 3 t 2 7. Check: d dx 3 2 5 x 5 y 2 1 C ± 5 x 3 y 2 y 5 2 5 x 5 y 2 1 C dy dx 5 x 3 y 2 Gi v en Re wr ite Inte g r a te Simplify 9. 3 4 x 4 y 3 1 C x 4 y 3 4 y 3 1 C E x 1 y 3 dx E 3 ! x dx 11. 2 2 ! x 1 C x 2 1 y 2 2 1 y 2 1 C E x 2 3 y 2 dx E 1 x ! x dx 13. 2 1 4 x 2 1 C 1 2 1 x 2 2 2 2 2 1 C 1 2 E x 2 3 dx E 1 2 x 3 dx 15. Check: d dx 3 x 2 2 1 3 x 1 C ± 5 x 1 3 E s x 1 3 d dx 5 x 2 2 1 3 x 1 C 17. Check: d dx f x 2 2 x 3 1 C g 5 2 x 2 3 x 2 E s 2 x 2 3 x 2 d dx 5 x 2 2 x 3 1 C 19. Check: d dx 1 1 4 x 4 1 2 x 1 C 2 5 x 3 1 2 E s x 3 1 2 d dx 5 1 4 x 4 1 2 x 1 C 21. Check: d dx 1 2 5 x 5 y 2 1 x 2 1 x 1 C 2 5 x 3 y 2 1 2 x 1 1 E s x 3 y 2 1 2 x 1 1 d dx 5 2 5 x 5 y 2 1 x 2 1 x 1 C 23. Check: d dx 1 3 5 x 5 y 3 1 C 2 5 x 2 y 3 5 3 ! x 2 E 3 ! x 2 dx 5 E x 2 y 3 dx 5 x 5 y 3 5 y 3 1 C 5 3 5 x 5 y 3 1 C 25. Check: d dx 1 2 1 2 x 2 1 C 2 5 1 x 3 E 1 x 3 dx 5 E x 2 3 dx 5 x 2 2 2 2 1 C 1 2 x 2 1 C
178 Chapter 4 Integration 27. Check: 5 x 2 1 x 1 1 ! x d dx 1 2 5 x 5 y 2 1 2 3 x 3 y 2 1 2 x 1 y 2 1 C 2 5 x 3 y 2 1 x 1 y 2 1 x 2 1 y 2 5 2 15 x 1 y 2 s 3 x 2 1 5 x 1 15 d 1 C 5 2 5 x 5 y 2 1 2 3 x 3 y 2 1 2 x 1 y 2 1 C E x 2 1 x 1 1 ! x dx 5 E s x 3 y 2 1 x 1 y 2 1 x 2 1 y 2 d dx 29. Check: 5 s x 1 1 ds 3 x 2 2 d d dx 1 x 3 1 1 2 x 2 2 2 x 1 C 2 5 3 x 2 1 x 2 2 5 x 3 1 1 2 x 2 2 2 x 1 C E s x 1 1 ds 3 x 2 2 d dx 5 E s 3 x 2 1 x 2 2 d dx 31. Check: d dy 1 2 7 y 7 y 2 1 C 2 5 y 5 y 2 5 y 2 ! y E y 2 ! y dy 5 E y 5 y 2 dy 5 2 7 y 7 y 2 1 C 33. Check: d dx s x 1 C d 5 1 E dx 5 E 1 dx 5 x 1 C 35. Check: d dx s 2 2 cos x 1 3 sin x 1 C d 5 2 sin x 1 3 cos x E s 2 sin x 1 3 cos x d dx 52 2 cos x 1 3 sin x 1 C 37. Check: d dt s t 1 csc t 1 C d 5 1 2 csc t cot t E s 1 2 csc t cot t d dt 5 t 1 csc t 1 C 39. Check: d d u s tan 1 cos 1 C d 5 sec 2 2 sin E s sec 2 2 sin d d 5 tan 1 cos 1 C 41. Check: d dy s tan y 1 C d 5 sec 2 y 5 tan 2 y 1 1 E s tan 2 y 1 1 d dy 5 E sec 2 y dy 5 tan y 1 C 43. 2 2 3 3 22 x 3 C 2 C 0 C y f s x d 5 cos x 45. Answers will vary. 5 4 3 3 2 1 2 3 y x 2 x ) x ) 2 2 x )) x f f f f s x d 5 2 x 1 C f 9 s x d 5 2 47. Answers will vary. 3 4 3 2 2 3 1 2 x y 3 3 3 x x x f x x 3 x ) f 3 ) f f s x d 5 x 2 x 3 3 1 C f s x d 5 1 2 x 2 49. y 5 x 2 2 x 1 1 1 5 s 1 d 2 2 s 1 d 1 C C 5 1 y 5 E s 2 x 2 1 d dx 5 x 2 2 x 1 C dy dx 5 2 x 2 1, s 1, 1 d

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Section 4.1 Antiderivatives and Indefinite Integration 179 51.
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ODD04 - CHAPTER Integration Section 4.1 Section 4.2 Section...

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