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Calculus, 2ADV C4 2019 HSC 13c

  1.  Differentiate  `(ln x)^2`.  (2 marks)

  2.  Hence, or otherwise, find  `int(ln x)/x\ dx`.  (1 mark)

Show Answers Only
  1. `(2 ln x)/x`
  2. `1/2 (ln x)^2 + C`
Show Worked Solution
i.    `y` `= (ln x)^2`
  `(dy)/(dx)` `= 2 ⋅ 1/x ⋅ ln x`
    `= (2 ln x)/x`

♦ Mean mark part (ii) 49%.

ii.    `int (ln x)/x\ dx` `=1/2 int (2 ln x)/x dx`
    `= 1/2 (ln x)^2 +C`

Calculus, 2ADV C4 2008 HSC 3b

  1. Differentiate  `log_e (cos x)`  with respect to  `x`.  (2 marks)

  2. Hence, or otherwise, evaluate  `int_0^(pi/4) tan x\ dx`.    (2 marks)

Show Answers Only
  1. `- tan x`
  2. `- log_e (1/sqrt2)\ \ text(or)\ \ 0.35\ \ text{(2 d.p.)}`
Show Worked Solution
i.    `y` `= log_e (cos x)`
  `dy/dx` `= (- sin x)/(cos x)`
    `= – tan x`

 

ii.    `int_0^(pi/4) tan x\ dx`
  `= – [log_e (cos x)]_0^(pi/4)`
  `= – [log_e(cos (pi/4)) – log_e (cos 0)]`
  `= – [log_e (1/sqrt2) – log_e 1]`
  `= – [log_e (1/sqrt2) – 0]`
  `= – log_e (1/sqrt2)`
  `= 0.346…`
  `= 0.35\ \ text{(2 d.p.)}`