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12 x1 t01 03 integrating derivative on function (2013)
 

12 x1 t01 03 integrating derivative on function (2013)

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