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Calculus, EXT1 C2 2020 HSC 13a

  1. Find  `d/(d theta) (sin^3 theta)`.  (1 mark)

  2. Use the substitution  `x = tan theta`  to evaluate  `int_0^1 (x^2)/(1 + x^2)^(5/2)\ dx`.  (4 marks)

Show Answers Only
  1. `3 cos theta sin^2 theta`
  2. `sqrt2/12`
Show Worked Solution

i.   `d/(d theta) (sin^3 theta) = 3 cos theta sin^2 theta`

 

ii.   `text(Let)\ x = tan theta`

`(dx)/(d theta) = sec^2 theta \ => \ dx  = sec^2 theta\ d theta`

`text(When)\ x = 1, \ theta = pi/4`

`text(When)\ x = 0, \ theta = 0`

`int_0^1 (x^2)/(1 + x^2)^(5/2) dx` `= int_0^(pi/4) (tan^2 theta)/((1 + tan^2 theta)^(5/2)) xx sec^2 theta\ d theta`
  `= int_0^(pi/4) (tan^2 theta)/((sec^2 theta)^(5/2)) xx sec^2 theta\ d theta`
  `= int_0^(pi/4) (sin^2 theta)/(cos^2 theta) · 1/((sec^2 theta)^(3/2))\ d theta`
  `= int_0^(pi/4) (sin^2 theta)/(cos^2 theta) · 1/(sec^3 theta)\ d theta`
  `= int_0^(pi/4) sin^2 theta cos theta\ d theta`
  `= 1/3[sin^3 theta]_0^(pi/4)`
  `= 1/3(sin^3\ pi/4 – 0)`
  `= 1/3 (1/sqrt2)^3`
  `= 1/(6sqrt2)`
  `= sqrt2/12`