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Calculus, EXT1 C2 2024 HSC 13b

  1. Show that  \(\cos ^4 x+\sin ^4 x=\dfrac{1+\cos ^2 2 x}{2}\).   (2 marks)

  2. Hence, or otherwise, evaluate  \(\displaystyle{\int}_0^{\frac{\pi}{4}}\left(\cos ^4 x+\sin ^4 x\right) d x\).  (3 marks)

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i.     \(\text{LHS}\) \(=\left[\dfrac{1}{2}(1+\cos (2 x)\right]^2+\left[\dfrac{1}{2}(1-\cos (2 x)\right]^2\)
    \(=\dfrac{1}{4}\left(1+2 \cos (2 x)+\cos ^2(2 x)+1-2 \cos (2 x)+\cos ^2(2 x)\right)\)
    \(=\dfrac{1}{4}\left(2+2 \cos ^{2}(2 x)\right)\)
    \(=\dfrac{1+\cos ^2(2 x)}{2}\)

  
ii.   \(\dfrac{3 \pi}{16}\)

Show Worked Solution

i.     \(\text{LHS}\) \(=\left[\dfrac{1}{2}(1+\cos (2 x)\right]^2+\left[\dfrac{1}{2}(1-\cos (2 x)\right]^2\)
    \(=\dfrac{1}{4}\left(1+2 \cos (2 x)+\cos ^2(2 x)+1-2 \cos (2 x)+\cos ^2(2 x)\right)\)
    \(=\dfrac{1}{4}\left(2+2 \cos ^{2}(2 x)\right)\)
    \(=\dfrac{1+\cos ^2(2 x)}{2}\)

  

ii.     \(\displaystyle{\int}_0^{\frac{\pi}{4}}\left(\cos ^4 x+\sin ^4 x\right) d x\)
    \(=\dfrac{1}{2} \displaystyle{\int}_0^{\frac{\pi}{4}} 1+\cos ^2(2 x) d x\)
    \(=\dfrac{1}{2} \displaystyle{\int}_0^{\frac{\pi}{4}} 1+\dfrac{1}{2}(1+\cos (4 x)) d x\)
    \(=\dfrac{1}{2}\left[\dfrac{3}{2}x +\dfrac{1}{8} \sin (4 x)\right]_0^{\frac{\pi}{4}}\)
    \(=\dfrac{1}{2}\left[\dfrac{3}{2} \times \dfrac{\pi}{4}+\dfrac{1}{8} \sin \pi-0\right]\)
    \(=\dfrac{3 \pi}{16}\)

Calculus, EXT1 C2 2010 HSC 2a

The derivative of a function  `f(x)`  is given by 

  `f^{′}(x) = sin^2 x`.

Find  `f(x)`, given that  `f(0) = 2`.   (2 marks)

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 `1/2x-1/4 sin 2x + 2`

Show Worked Solution
`f^{′}(x)` `= sin^2 x`
`f(x)` `= int sin^2 x\ dx`
  `= int 1/2(1-cos 2x)\  dx`
  `= int 1/2-1/2 cos 2x\  dx`
  `= 1/2 x-1/4 sin 2 x + c`

 
`text(Given)\ \ f(0) = 2,`

`2` `= 1/2 xx 0-1/4 sin 0 + c`
 `:. c= 2`

 
`:.\ f(x) = 1/2 x-1/4 sin 2x + 2`

Calculus, EXT1 C2 2012 HSC 7 MC

Which expression is equal to `int sin^2 3x\ dx`?

  1. `1/2 (x-1/3 sin 3x) + C`
  2. `1/2 (x + 1/3 sin 3x) + C`
  3. `1/2 (x-1/6 sin 6x) + C`
  4. `1/2 (x + 1/6 sin 6x) + C`
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`C`

Show Worked Solution

`text(Using:)\ \ sin^2a = 1/2 (1-cos 2a)`

`int sin^2 3x\ dx` `= 1/2 int (1-cos 6x)\ dx`
  `= 1/2 (x-1/6 sin 6x) + C`

`=>  C`