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Calculus, EXT2 C1 2016 HSC 12b

  1. Differentiate  `x\ f(x)-int x\ f^(′)(x)\ dx.`  (1 mark)

  2. Hence, or otherwise, find  `int tan^-1 x\ dx.`  (2 marks)

Show Answers Only
  1. `f(x)`
  2. `x tan^(−1)x-1/2 ln(1 + x^2)`
Show Worked Solution

i.   `d/dx (x\ f(x)-int x\ f^(′)(x)\ dx)`

`= x\ f^(′)(x) + f(x)-x\ fprime(x)`

`= f(x)`

 

ii.    `int tan^(−1)x\ dx` `= x\ tan^(−1)x-int x/(1 + x^2)\ dx`
    `= x\ tan^(−1)x-1/2 ln(1 + x^2)+c`