12 x1 t01 03 integrating derivative on function (2013)

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12 x1 t01 03 integrating derivative on function (2013)

  1. 1. Integrating Derivative on Function
  2. 2. Integrating Derivative on Function  f  x  dx  log f  x   c f x
  3. 3. Integrating Derivative on Function  e.g. (i) 1  7  3x dx f  x  dx  log f  x   c f x
  4. 4. Integrating Derivative on Function  e.g. (i) 1  7  3x dx 1 3   dx 3 7  3x f  x  dx  log f  x   c f x
  5. 5. Integrating Derivative on Function  e.g. (i) f  x  dx  log f  x   c f x 1  7  3x dx 1 3   dx 3 7  3x 1   log7  3 x   c 3
  6. 6. Integrating Derivative on Function  e.g. (i) f  x  dx  log f  x   c f x 1  7  3x dx 1 3   dx 3 7  3x 1   log7  3 x   c 3 ii  dx  8x  5
  7. 7. Integrating Derivative on Function  e.g. (i) f  x  dx  log f  x   c f x 1  7  3x dx 1 3   dx 3 7  3x 1   log7  3 x   c 3 dx  8x  5 1 8dx   8 8x  5 ii 
  8. 8. Integrating Derivative on Function  e.g. (i) f  x  dx  log f  x   c f x 1  7  3x dx 1 3   dx 3 7  3x 1   log7  3 x   c 3 dx  8x  5 1 8dx   8 8x  5 1  log8 x  5  c 8 ii 
  9. 9. Integrating Derivative on Function  f  x  dx  log f  x   c f x e.g. (i)  1 dx 7  3x 1 3   dx 3 7  3x 1   log7  3 x   c 3 dx ii   8x  5 1 8dx   8 8x  5 1  log8 x  5  c 8 x5 iii   6 dx x 2
  10. 10. Integrating Derivative on Function  f  x  dx  log f  x   c f x e.g. (i)  1 dx 7  3x 1 3   dx 3 7  3x 1   log7  3 x   c 3 dx ii   8x  5 1 8dx   8 8x  5 1  log8 x  5  c 8 x5 iii   6 dx x 2 1 6 x5   6 dx 6 x 2
  11. 11. Integrating Derivative on Function  f  x  dx  log f  x   c f x e.g. (i)  1 dx 7  3x 1 3   dx 3 7  3x 1   log7  3 x   c 3 dx ii   8x  5 1 8dx   8 8x  5 1  log8 x  5  c 8 x5 iii   6 dx x 2 1 6 x5   6 dx 6 x 2 1  logx 6  2   c 6
  12. 12. 1 iv   dx 5x
  13. 13. 1 iv   dx 5x 1 5   dx 5 5x
  14. 14. 1 iv   dx 5x 1 5   dx 5 5x 1  log 5 x  c 5
  15. 15. 1 iv   dx 5x 1 5   dx 5 5x 1  log 5 x  c 5 OR 1 1  x dx 5
  16. 16. 1 iv   dx 5x 1 5   dx 5 5x 1  log 5 x  c 5 OR 1 1  x dx 5 1  log x  c 5
  17. 17. 1 iv   dx 5x 1 5   dx 5 5x 1  log 5 x  c 5 4x 1 v   dx 2x 1 OR 1 1  x dx 5 1  log x  c 5
  18. 18. 1 iv   dx 5x 1 5   dx 5 5x 1  log 5 x  c 5 4x 1 v   dx 2x 1 OR 1 1  x dx 5 1  log x  c 5 order numerator  order denominator
  19. 19. 1 iv   dx 5x 1 5   dx 5 5x 1  log 5 x  c 5 4x 1 v   dx 2x 1 OR 1 1  x dx 5 1  log x  c 5 order numerator  order denominator  polynomial division
  20. 20. 1 iv   dx 5x 1 5   dx 5 5x 1  log 5 x  c 5 4x 1 v   dx 2x 1 OR 1 1  x dx 5 1  log x  c 5 order numerator  order denominator  polynomial division 2 2x 1 4x 1 4x  2 1
  21. 21. 1 iv   dx 5x 1 5   dx 5 5x 1  log 5 x  c 5 4x 1 dx v   2x 1 2  1  dx     2 x  1 OR 1 1  x dx 5 1  log x  c 5 order numerator  order denominator  polynomial division 2 2x 1 4x 1 4x  2 1
  22. 22. 1 iv   dx 5x 1 5   dx 5 5x 1  log 5 x  c 5 OR 1 1  x dx 5 1  log x  c 5 order numerator  order denominator 4x 1 dx v    polynomial division 2x 1 2  1  dx 2   2x 1 4x 1   2 x  1 1 4x  2  2 x  log2 x  1  c 2 1
  23. 23. vi  2 2x  x 2  1dx 1
  24. 24. vi  2 2x  x 2  1dx 1  logx  11 2 2
  25. 25. vi  2 2x  x 2  1dx 1  logx  11 2 2  log 5  log 2
  26. 26. vi  2 2x  x 2  1dx 1  logx  11 2 2  log 5  log 2 5  log  2
  27. 27. vi  2 vii  Differentiate x 3 log x and hence 2x  x 2  1dx 1  logx  11 2 2  log 5  log 2 5  log  2 integrate x 2 log x
  28. 28. vi  2 vii  Differentiate x 3 log x and hence 2x  x 2  1dx 1  logx  11 2 2  log 5  log 2 5  log  2 integrate x 2 log x d 3 3 1 x log x  x    log x 3x 2  dx  x
  29. 29. vi  2 vii  Differentiate x 3 log x and hence 2x  x 2  1dx 1  logx  11 2 2  log 5  log 2 5  log  2 integrate x 2 log x d 3 3 1 x log x  x    log x 3x 2  dx  x  x 2  3 x 2 log x
  30. 30. vi  2 vii  Differentiate x 3 log x and hence 2x  x 2  1dx 1  logx  11 2 2  log 5  log 2 5  log  2 integrate x 2 log x d 3 3 1 x log x  x    log x 3x 2  dx  x  x 2  3 x 2 log x  x 2  3 x 2 log x dx  x 3 log x  c
  31. 31. vi  2 vii  Differentiate x 3 log x and hence 2x  x 2  1dx 1  logx  11 2 2  log 5  log 2 5  log  2 integrate x 2 log x d 3 3 1 x log x  x    log x 3x 2  dx  x  x 2  3 x 2 log x  x 2  3 x 2 log x dx  x 3 log x  c 3 x 2 log xdx  x 3 log x   x 2 dx  c
  32. 32. vi  2 vii  Differentiate x 3 log x and hence 2x  x 2  1dx 1  logx  11 2 2  log 5  log 2 5  log  2 integrate x 2 log x d 3 3 1 x log x  x    log x 3x 2  dx  x  x 2  3 x 2 log x  x 2  3 x 2 log x dx  x 3 log x  c 3 x 2 log xdx  x 3 log x   x 2 dx  c 1 3 1 3  x log xdx  3 x log x  9 x  c 2
  33. 33. vi  2 vii  Differentiate x 3 log x and hence 2x  x 2  1dx 1  logx  11 2 2  log 5  log 2 5  log  2 integrate x 2 log x d 3 3 1 x log x  x    log x 3x 2  dx  x  x 2  3 x 2 log x  x 2  3 x 2 log x dx  x 3 log x  c 3 x 2 log xdx  x 3 log x   x 2 dx  c 1 3 1 3  x log xdx  3 x log x  9 x  c 2 Exercise 12D; 1 to 12 ace in all, 14a* Exercise 12E; 1 to 6 all, 7 to 21 odds, 22abc*, 23*

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