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12X1 T01 03 integrating derivative on function (2010)

N
Nigel Simmons

The document discusses various techniques for integrating derivatives. It provides examples of integrating common derivatives and explains the steps. Example integrals presented include 1/(7-3x), 8x+5, x5, 1/5x, (4x+1)/(2x+1), ∫x2/(x2+1)dx, and differentiating and integrating x3logx.

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Integrating Derivative on Function
Integrating Derivative on Function f  x   f x dx  log f  x   c
Integrating Derivative on Function f  x   f x dx  log f  x   c 1 e.g. (i)  7  3x dx
Integrating Derivative on Function f  x   f x dx  log f  x   c 1 e.g. (i)  7  3x dx 1 3   dx 3 7  3x
Integrating Derivative on Function f  x   f x dx  log f  x   c 1 e.g. (i)  7  3x dx 1 3   dx 3 7  3x 1   log7  3 x   c 3
Integrating Derivative on Function f  x   f x dx  log f  x   c 1 dx e.g. (i)  7  3x dx ii   8x  5 1 3   dx 3 7  3x 1   log7  3 x   c 3
Integrating Derivative on Function f  x   f x dx  log f  x   c 1 dx e.g. (i)  7  3x dx ii  8x  5 1 3 1 8dx   dx   3 7  3x 8 8x  5 1   log7  3 x   c 3
Integrating Derivative on Function f  x   f x dx  log f  x   c 1 dx e.g. (i)  7  3x dx ii  8x  5 1 3 1 8dx   dx   3 7  3x 8 8x  5 1 1   log7  3 x   c  log8 x  5  c 3 8
Integrating Derivative on Function f  x   f x dx  log f  x   c e.g. (i)  1 dx ii   dx x5 iii   6 dx 7  3x 8x  5 x 2 1 3 1 8dx   dx   3 7  3x 8 8x  5 1 1   log7  3 x   c  log8 x  5  c 3 8
Integrating Derivative on Function f  x   f x dx  log f  x   c e.g. (i)  1 dx ii   dx x5 iii   6 dx 7  3x 8x  5 x 2 1 3 1 8dx 1 6 x5   dx     6 dx 3 7  3x 8 8x  5 6 x 2 1 1   log7  3 x   c  log8 x  5  c 3 8
Integrating Derivative on Function f  x   f x dx  log f  x   c e.g. (i)  1 dx ii   dx x5 iii   6 dx 7  3x 8x  5 x 2 1 3 1 8dx 1 6 x5   dx     6 dx 3 7  3x 8 8x  5 6 x 2  logx 6  2   c 1 1 1   log7  3 x   c  log8 x  5  c 3 8 6
1 iv   dx 5x
1 iv   dx 5x 1 5   dx 5 5x
1 iv   dx 5x 1 5   dx 5 5x 1  log 5 x  c 5
1 iv   dx OR 1 1  x dx 5x 5 1 5   dx 5 5x 1  log 5 x  c 5
1 iv   dx OR 1 1  x dx 5x 5 1 5   dx 1  log x  c 5 5x 5 1  log 5 x  c 5
1 iv   dx OR 1 1  x dx 5x 5 1 5   dx 1  log x  c 5 5x 5 1  log 5 x  c 5 4x 1 v   dx 2x 1
1 iv   dx OR 1 1  x dx 5x 5 1 5   dx 1  log x  c 5 5x 5 1  log 5 x  c 5 4x 1 order numerator  order denominator v   dx 2x 1
1 iv   dx OR 1 1  x dx 5x 5 1 5   dx 1  log x  c 5 5x 5 1  log 5 x  c 5 4x 1 order numerator  order denominator v   dx  polynomial division 2x 1
1 iv   dx OR 1 1  x dx 5x 5 1 5   dx 1  log x  c 5 5x 5 1  log 5 x  c 5 4x 1 order numerator  order denominator v   dx  polynomial division 2x 1 2 2x 1 4x 1 4x  2 1
1 iv   dx OR 1 1  x dx 5x 5 1 5   dx 1  log x  c 5 5x 5 1  log 5 x  c 5 4x 1 order numerator  order denominator v   dx  polynomial division 2x 1  2  1  dx 2  2 x  1 2x 1 4x 1 4x  2 1
1 iv   dx OR 1 1  x dx 5x 5 1 5   dx 1  log x  c 5 5x 5 1  log 5 x  c 5 4x 1 order numerator  order denominator v   dx  polynomial division 2x 1  2  1  dx 2  2 x  1 2x 1 4x 1 1 4x  2  2 x  log2 x  1  c 2 1
2 2x vi   x 2  1dx 1
2 2x vi   x 2  1dx 1  logx  11 2 2
2 2x vi   x 2  1dx 1  logx  11 2 2  log 5  log 2
2 2x vi   x 2  1dx 1  logx  11 2 2  log 5  log 2  log  5   2
2 vi  2x  x 2  1dx vii  Differentiate x 3 log x and hence 1 integrate x 2 log x  logx  11 2 2  log 5  log 2  log  5   2
2 vi  2x  x 2  1dx vii  Differentiate x 3 log x and hence 1 integrate x 2 log x  logx  11 2 x log x  x    log x 3x 2  3 1 2 d 3 dx  x  log 5  log 2  log  5   2
2 vi  2x  x 2  1dx vii  Differentiate x 3 log x and hence 1 integrate x 2 log x  logx  11 2 x log x  x    log x 3x 2  3 1 2 d 3 dx  x  log 5  log 2  x 2  3 x 2 log x  log  5   2
2 vi  2x  x 2  1dx vii  Differentiate x 3 log x and hence 1 integrate x 2 log x  logx  11 2 x log x  x    log x 3x 2  3 1 2 d 3 dx  x  log 5  log 2  x 2  3 x 2 log x  log  5  x  3 x 2 log x dx  x 3 log x  c   2 2
2 vi  2x  x 2  1dx vii  Differentiate x 3 log x and hence 1 integrate x 2 log x  logx  11 2 x log x  x    log x 3x 2  3 1 2 d 3 dx  x  log 5  log 2  x 2  3 x 2 log x  log  5  x  3 x 2 log x dx  x 3 log x  c   2 2 3 x 2 log xdx  x 3 log x   x 2 dx  c
2 vi  2x  x 2  1dx vii  Differentiate x 3 log x and hence 1 integrate x 2 log x  logx  11 2 x log x  x    log x 3x 2  3 1 2 d 3 dx  x  log 5  log 2  x 2  3 x 2 log x  log  5  x  3 x 2 log x dx  x 3 log x  c   2 2 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
2 vi  2x  x 2  1dx vii  Differentiate x 3 log x and hence 1 integrate x 2 log x  logx  11 2 x log x  x    log x 3x 2  3 1 2 d 3 dx  x  log 5  log 2  x 2  3 x 2 log x  log  5  x  3 x 2 log x dx  x 3 log x  c   2 2 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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12X1 T01 03 integrating derivative on function (2010)

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