Exercise 3.3
Exercise 3.3 Exercise 3.3
MA111: Prepared by Dr.Archara Pacheenburawana 45 Answer to Exercise 5.3 1. (a) 2π (b) π(1−1/e) (c) 64π/15 2. (a) 21π/2 (b) 1944π/5 (c) 5π/6 3. (a) 36√ 3 5 π (b) 40√ 5 3 π (c) 23π 30 (d) π 2 (e) 8π 3 Find the length of the curve. Exercise 5.4 1. y = 1 3 (x2 +2) 3/2 , 0 ≤ x ≤ 1 2. y = x4 4 + 1 8x 2, 1 ≤ x ≤ 3 3. y = ln(secx), 0 ≤ x ≤ π/4 4. y = ln(1−x 2 ), 0 ≤ x ≤ 1 2 5. y = coshx, 0 ≤ x ≤ 1 Answer to Exercise 5.4 1. 4 3 2. 181 9 3. ln( √ 2+1) 4. ln3− 1 2 5. sinh1 Exercise 6.1 Evaluate the integral. ∫ ∫ 1. xe 2x dx 2. x 2 lnxdx ∫ ∫ 3. xsin4xdx 4. x 2 e −3x dx ∫ ∫ 5. x 2 cos3xdx 6. e x sin4xdx ∫ ∫ 7. (lnx) 2 dx 8. cosxcos2xdx ∫ ∫ 9. xsec 2 xdx 10. cosxln(sinx)dx ∫ ∫ 11. cos(lnx)dx 12. cos −1 xdx ∫ 13. sin √ ∫ 1 xdx 14. xsin2xdx 15. 17. ∫ 1 0 ∫ 4 1 xe −x dx 16. ln √ xdx 18. 0 ∫ 2 1 ∫ 2 1 lnx x 2 dx x 4 (lnx) 2 dx
MA111: Prepared by Dr.Archara Pacheenburawana 46 Answer to Exercise 6.1 1. 1 2 xe2x − 1 4 e2x +C 2. 1 3 x2 lnx− 1 9 x3 +C 3. − 1 4 xcos4x+ 1 16 sin4x+C 4. − 1 3 x2 e −3x − 2 9 xe−3x − 2 27 e−3x +C 5. 1 3 x2 cos3x+ 2 9 xcos3x− 2 27 sin3x+C 6. 1 17 ex sin4x− 4 17 ex cos4x+C 7. x(lnx) 2 −2xlnx+2x+C 8. 2 sin2xcosx− 1 cos2xsinx+C 9. xtanx+ln|cosx|+C 3 3 10. sinxln(sinx)−sinx+C 11. 1 2 x[sin(lnx)+cos(lnx)]+C 12. xcos −1 x− √ 1−x 2 +C 13. −2 √ xcos √ x+2sin √ x+C 14. 1 4 sin2− 1 2 cos2 15. 1− 2 16. 1 − 1 ln2 17. 2ln4− 3 18. 3 e 2 2 2 2 (ln2)2 − 64 62 ln2+ 25 125 Exercise 6.2 Evaluate the integral. ∫ ∫ 1. cosxsin 4 xdx 2. cos 2 xsinxdx ∫ 3π/4 ∫ 3. sin 5 xcos 3 xdx 4. cos 5 xsin 4 xdx π/2 ∫ π/2 ∫ 5. sin 2 3xdx 6. cos 2 xsin 2 xdx 0 ∫ ∫ π/4 7. (1−sin2x) 2 dx 8. sin 4 xcos 2 xdx 0 ∫ 9. sin 3 x √ ∫ cosxdx 10. cos 2 xtan 3 xdx ∫ ∫ 1−sinx 11. cosx dx 12. tan 2 xdx ∫ ∫ 13. sec 4 xdx 14. tan 5 xdx ∫ ∫ π/4 15. tanxsec 3 xdx 16. tan 4 xsec 2 xdx ∫ 17. tan 3 xsecxdx 18. 0 ∫ π/3 0 tan 3 xsecxdx ∫ sec 2 ∫ x π/2 19. cotx dx 20. cot 2 xdx π/6 ∫ ∫ 21. cot 3 xcsc 3 xdx 22. cot 2 ωcsc 4 ωdω
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MA111: Prepared by Dr.Archara Pacheenburawana 46<br />
Answer to <strong>Exercise</strong> 6.1<br />
1.<br />
1<br />
2 xe2x − 1 4 e2x +C 2. 1 3 x2 lnx− 1 9 x3 +C 3. − 1 4 xcos4x+ 1 16 sin4x+C<br />
4. − 1 3 x2 e −3x − 2 9 xe−3x − 2 27 e−3x +C 5. 1 3 x2 cos3x+ 2 9 xcos3x− 2 27 sin3x+C<br />
6.<br />
1<br />
17 ex sin4x− 4<br />
17 ex cos4x+C 7. x(lnx) 2 −2xlnx+2x+C<br />
8.<br />
2<br />
sin2xcosx− 1 cos2xsinx+C 9. xtanx+ln|cosx|+C<br />
3 3<br />
10. sinxln(sinx)−sinx+C 11. 1 2 x[sin(lnx)+cos(lnx)]+C<br />
12. xcos −1 x− √ 1−x 2 +C 13. −2 √ xcos √ x+2sin √ x+C 14. 1 4 sin2− 1 2 cos2<br />
15. 1− 2 16. 1 − 1 ln2 17. 2ln4− 3 18. 3 e 2 2 2 2 (ln2)2 − 64 62<br />
ln2+<br />
25 125<br />
<strong>Exercise</strong> 6.2<br />
Evaluate the integral.<br />
∫<br />
∫<br />
1. cosxsin 4 xdx 2. cos 2 xsinxdx<br />
∫ 3π/4<br />
∫<br />
3. sin 5 xcos 3 xdx 4. cos 5 xsin 4 xdx<br />
π/2<br />
∫ π/2<br />
∫<br />
5. sin 2 3xdx 6. cos 2 xsin 2 xdx<br />
0<br />
∫<br />
∫ π/4<br />
7. (1−sin2x) 2 dx 8. sin 4 xcos 2 xdx<br />
0<br />
∫<br />
9. sin 3 x √ ∫<br />
cosxdx 10. cos 2 xtan 3 xdx<br />
∫ ∫<br />
1−sinx<br />
11.<br />
cosx dx<br />
12. tan 2 xdx<br />
∫<br />
∫<br />
13. sec 4 xdx 14. tan 5 xdx<br />
∫<br />
∫ π/4<br />
15. tanxsec 3 xdx 16. tan 4 xsec 2 xdx<br />
∫<br />
17. tan 3 xsecxdx 18.<br />
0<br />
∫ π/3<br />
0<br />
tan 3 xsecxdx<br />
∫ sec 2 ∫<br />
x<br />
π/2<br />
19.<br />
cotx dx 20. cot 2 xdx<br />
π/6<br />
∫<br />
∫<br />
21. cot 3 xcsc 3 xdx 22. cot 2 ωcsc 4 ωdω
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