12 x1 t02 02 integrating exponentials (2014)

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12 x1 t02 02 integrating exponentials (2014)

  1. 1. Integrating Exponentials
  2. 2. Integrating Exponentials e ax dx  
  3. 3. Integrating Exponentials 1 ax  e dx  a e  c ax
  4. 4. Integrating Exponentials 1 ax  e dx  a e  c ax  f  x e f  x dx 
  5. 5. Integrating Exponentials 1 ax  e dx  a e  c ax  f  x e f  x dx  e f  x   c
  6. 6. Integrating Exponentials 1 ax  e dx  a e  c ax  e.g. i   e5 x dx f  x e f  x dx  e f  x   c
  7. 7. Integrating Exponentials 1 ax  e dx  a e  c ax  e.g. i   e5 x dx 1  e5 x  c 5 f  x e f  x dx  e f  x   c
  8. 8. Integrating Exponentials 1 ax  e dx  a e  c ax  e.g. i   e5 x dx 1  e5 x  c 5 f  x e f  x dx  e f  x   c OR e5 x dx  1   5e5 x dx 5
  9. 9. Integrating Exponentials 1 ax  e dx  a e  c ax  e.g. i   e5 x dx 1  e5 x  c 5 f  x e f  x dx  e f  x   c OR e5 x dx  1   5e5 x dx 5 1  e5 x  c 5
  10. 10. ii   xe dx x2
  11. 11. 1 x2 ii   xe dx   2 xe dx 2 x2
  12. 12. 1 x2 ii   xe dx   2 xe dx 2 1 x2  e c 2 x2
  13. 13. 1 x2 ii   xe dx   2 xe dx 2 1 x2  e c 2 x2 iii   e9 x 5 dx  1  9e9 x 5 dx 9
  14. 14. 1 x2 ii   xe dx   2 xe dx 2 1 x2  e c 2 x2 iii   e9 x 5 dx  1  9e9 x 5 dx 9 1 9 x 5  e c 9
  15. 15. 1 x2 ii   xe dx   2 xe dx 2 1 x2  e c 2 x2 iv   e x dx iii   e9 x 5 dx  1  9e9 x 5 dx 9 1 9 x 5  e c 9
  16. 16. 1 x2 ii   xe dx   2 xe dx 2 1 x2  e c 2 x2 iv   x 2 e x dx   e dx iii   e9 x 5 dx  1  9e9 x 5 dx 9 1 9 x 5  e c 9
  17. 17. 1 x2 ii   xe dx   2 xe dx 2 1 x2  e c 2 x2 iv   x 2 e x dx   e dx x 1  2  e 2 dx 2 iii   e9 x 5 dx  1  9e9 x 5 dx 9 1 9 x 5  e c 9
  18. 18. 1 x2 ii   xe dx   2 xe dx 2 1 x2  e c 2 x2 iv   x 2 e x dx   e dx x 1  2  e 2 dx 2 x 2  2e  c  2 ex  c iii   e9 x 5 dx  1  9e9 x 5 dx 9 1 9 x 5  e c 9
  19. 19. 1 x2 ii   xe dx   2 xe dx 2 1 x2  e c 2 x2 iv   x 2 e x dx   e dx x 1  2  e 2 dx 2 x 2  2e  c  2 ex  c iii   e9 x 5 dx  1  9e9 x 5 dx 9 1 9 x 5  e c 9 v   e x  1e x  3dx
  20. 20. 1 x2 ii   xe dx   2 xe dx 2 1 x2  e c 2 x2 iv   x 2 e x dx   e dx x 1  2  e 2 dx 2 x 2  2e  c  2 ex  c iii   e9 x 5 dx  1  9e9 x 5 dx 9 1 9 x 5  e c 9 v   e  1e  3dx   e  2e  3dx x x 2x x
  21. 21. 1 x2 ii   xe dx   2 xe dx 2 1 x2  e c 2 x2 iv   x 2 e x dx   e dx x 1  2  e 2 dx 2 x 2  2e  c  2 ex  c iii   e9 x 5 dx  1  9e9 x 5 dx 9 1 9 x 5  e c 9 v   e  1e  3dx   e  2e  3dx x x 2x x 1 2x  e  2e x  3 x  c 2
  22. 22. 1 x2 ii   xe dx   2 xe dx 2 1 x2  e c 2 x2 iv   x 2 e x dx   e dx x 1  2  e 2 dx 2 x 2  2e  c  2 ex  c iii   e9 x 5 dx  1  9e9 x 5 dx 9 1 9 x 5  e c 9 v   e  1e  3dx   e  2e  3dx x x 2x x 1 2x  e  2e x  3 x  c 2 e5 x  e x vi   2 x dx e
  23. 23. 1 x2 ii   xe dx   2 xe dx 2 1 x2  e c 2 x2 iv   x 2 e x dx   e dx x 1  2  e 2 dx 2 x 2  2e  c  2 ex  c iii   e9 x 5 dx  1  9e9 x 5 dx 9 1 9 x 5  e c 9 v   e  1e  3dx   e  2e  3dx x x 2x x 1 2x  e  2e x  3 x  c 2 e5 x  e x vi   2 x dx e   e3 x  e  x dx
  24. 24. 1 x2 ii   xe dx   2 xe dx 2 1 x2  e c 2 x2 iv   x 2 e x dx   e dx x 1  2  e 2 dx 2 x 2  2e  c  2 ex  c iii   e9 x 5 dx  1  9e9 x 5 dx 9 1 9 x 5  e c 9 v   e  1e  3dx   e  2e  3dx x x 2x x 1 2x  e  2e x  3 x  c 2 e5 x  e x vi   2 x dx e   e3 x  e  x dx 1 3x x  e e c 3
  25. 25. vii  1 x e 2 x3 0 dx
  26. 26. vii  1 x e 2 x3 0 1 dx 1 2 x3   3 x e dx 30
  27. 27. vii  1 x e 2 x3 0 1 dx 1 2 x3   3 x e dx 30 1 x3 1  e 0 3  
  28. 28. vii  1 x e 2 x3 0 1 dx 1 2 x3   3 x e dx 30 1 x3 1  e 0 3 1 1 0  e  e  3  
  29. 29. vii  1 x e 2 x3 0 1 dx 1 2 x3   3 x e dx 30 1 x3 1  e 0 3 1 1 0  e  e  3 1  e  1 3  
  30. 30. vii  1 x e 2 x3 0 1 dx 1 2 x3   3 x e dx 30 1 x3 1  e 0 3 1 1 0  e  e  3 1  e  1 3   viii   3x dx
  31. 31. vii  1 x e 2 x3 0 1 dx 1 2 x3   3 x e dx 30 1 x3 1  e 0 3 1 1 0  e  e  3 1  e  1 3   viii   3x dx 3x  c log 3
  32. 32. vii  1 x e 2 x3 0 1 dx 1 2 x3   3 x e dx 30 1 x3 1  e 0 3 1 1 0  e  e  3 1  e  1 3 viii   3x dx 3x  c log 3   Exercise 13C; 2 to 8 ace etc, 9, 10, 11, 13, 17 Exercise 13D; 2 to 18 evens, 21*

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