Arc length with inverse trig

A question is this type if and only if it requires finding arc length of a curve where the resulting integral involves inverse trigonometric or hyperbolic functions.

2 questions · Challenging +1.3

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Edexcel FP3 2018 June Q4
12 marks Challenging +1.3
4. The curve \(C\) has equation $$y = \operatorname { arsinh } x + x \sqrt { x ^ { 2 } + 1 } , \quad 0 \leqslant x \leqslant 1$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \sqrt { x ^ { 2 } + 1 }\)
  2. Hence show that the length of the curve \(C\) is given by $$\int _ { 0 } ^ { 1 } \sqrt { 4 x ^ { 2 } + 5 } d x$$
  3. Using the substitution \(x = \frac { \sqrt { 5 } } { 2 } \sinh u\), find the exact length of the curve \(C\), giving your answer in the form \(a + b \ln c\), where \(a , b\) and \(c\) are constants to be found.
AQA FP2 2016 June Q3
10 marks Challenging +1.3
3 The arc of the curve with equation \(y = 4 - \ln \left( 1 - x ^ { 2 } \right)\) from \(x = 0\) to \(x = \frac { 3 } { 4 }\) has length \(s\).
  1. Show that \(s = \int _ { 0 } ^ { \frac { 3 } { 4 } } \left( \frac { 1 + x ^ { 2 } } { 1 - x ^ { 2 } } \right) \mathrm { d } x\).
  2. Find the value of \(s\), giving your answer in the form \(p + \ln N\), where \(p\) is a rational number and \(N\) is an integer.
    [0pt] [6 marks]
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