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Calculus, EXT1 C2 EQ-Bank 4

Find the value of \(n\), given

\(\displaystyle \int_0^n-\dfrac{1}{\sqrt{1-x^2}}=-\dfrac{\pi}{6}\).   (2 marks)

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\(n=\dfrac{1}{2}\)

Show Worked Solution
\(\displaystyle\int_0^n-\dfrac{1}{\sqrt{1-x^2}}\) \(=-\dfrac{\pi}{6}\)
\(\left[\cos ^{-1} x\right]_0^n\) \(=-\dfrac{\pi}{6}\)
\(\cos ^{-1}(n)-\cos ^{-1} 0\) \(=-\dfrac{\pi}{6}\)
\(\cos ^{-1}(n)-\dfrac{\pi}{2}\) \(=-\dfrac{\pi}{6}\)
\(\cos ^{-1}(n)\) \(=\dfrac{\pi}{2}-\dfrac{\pi}{6}\)
\(n\) \(=\cos \left(\dfrac{\pi}{3}\right)\)
  \(=\dfrac{1}{2}\)

Calculus, EXT2 C2 EQ-Bank 3

  1. Find  \(\displaystyle \int \frac{1}{\sqrt{4 x-x^2}}\, d x\).   (2 marks)

  2. Determine the values of \(x\) for which the antiderivative  \(\int \dfrac{1}{\sqrt{4 x-x^2}}\, d x\)  is real and finite.     (1 mark)

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a.   \(\sin ^{-1}\left(\frac{x-2}{2}\right)+c\)

b.  \(0<x<4\)

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a.     \(\displaystyle\int \frac{1}{\sqrt{4 x-x^2}}\, d x\) \(=\displaystyle\int \frac{1}{\sqrt{4-4+4 x-x^2}}\, d x\)
    \(=\displaystyle\int \frac{1}{\sqrt{4-(x-2)^2}}\, d x\)
    \(=\sin ^{-1}\left(\dfrac{x-2}{2}\right)+c\)

 

b.    \(4 x-x^2>0\)

\(x(4-x)>0\)

\(0<x<4\)

Calculus, EXT1 C2 EQ-Bank 2

Find  \(\displaystyle \int \frac{1}{\sqrt{15-2 x-x^2}}\, d x\).   (3 marks)

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\(\displaystyle\sin ^{-1}\left(\frac{x+1}{4}\right)+c\)

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\(\displaystyle\int \frac{1}{\sqrt{15-2 x-x^2}}\, d x\) \(=\displaystyle\int \frac{1}{\sqrt{16-1-2 x-x^2}}\, d x\)
  \(=\displaystyle\int \frac{1}{\sqrt{16-(x+1)^2}}\, d x\)
  \(=\displaystyle\sin ^{-1}\left(\frac{x+1}{4}\right)+c\)

Calculus, EXT1 C2 EQ-Bank 1 MC

\(\displaystyle \int \dfrac{1}{\sqrt{5-4 x-x^2}}\, d x=\)
 

  1. \(\sin ^{-1}\left(\dfrac{x+2}{3}\right)+c \)
  2. \( \sin ^{-1}\left(\dfrac{x-2}{3}\right)+c \)
  3. \(\sin ^{-1}\left(\dfrac{x+2}{9}\right)+c \)
  4. \(\sin ^{-1}\left(\dfrac{x-2}{9}\right)+c\)
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\(\Rightarrow A\)

Show Worked Solution
\(\displaystyle \int \frac{1}{\sqrt{5-4 x-x^2}}\, d x\) \(=\displaystyle\int \frac{1}{\sqrt{9-4-4 x-x^2}}\,dx\)
  \(=\displaystyle \int \frac{1}{\sqrt{9-(x+2)^2}}\,dx\)
  \(=\sin ^{-1}\left(\dfrac{x+2}{3}\right)+c\)

\(\Rightarrow A\)

Calculus, EXT1 C2 2015 HSC 7 MC

What is the value of `k` such that `int_0^k 1/sqrt(4 − x^2) \ dx= pi/3 ?`

  1. `1`
  2. `sqrt3`
  3. `2`
  4. `2sqrt3`
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`B`

Show Worked Solution
`int_0^1 1/sqrt(4 − x^2)dx` `= pi/3`
`[sin^(−1)\ x/2]_0^k` `= pi/3`
`sin^(−1)\ k/2 − sin^(−1)\ 0` `= pi/3`
`sin^(−1)\ k/2` `= pi/3`
`k/2` `= sin\ pi/3`
  `= sqrt3/2`
`:.k` `= sqrt3`

`⇒ B`