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

This preview shows pages 1–5. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Section 4.1 Antiderivatives and Indefinite Integration 179 51.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern