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# 12 x1 t01 03 integrating derivative on function (2013)

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## 12 x1 t01 03 integrating derivative on function (2013)Presentation Transcript

• Integrating Derivative on Function
• Integrating Derivative on Function  f  x  dx  log f  x   c f x
• Integrating Derivative on Function  e.g. (i) 1  7  3x dx f  x  dx  log f  x   c f x
• 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
• 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
• 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
• 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 
• 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 
• 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
• 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
• 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
• 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 5x 1 5   dx 5 5x 1  log 5 x  c 5 OR 1 1  x dx 5
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• vi  2 2x  x 2  1dx 1
• vi  2 2x  x 2  1dx 1  logx  11 2 2
• vi  2 2x  x 2  1dx 1  logx  11 2 2  log 5  log 2
• vi  2 2x  x 2  1dx 1  logx  11 2 2  log 5  log 2 5  log  2
• 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
• 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
• 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
• 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
• 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
• 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
• 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*