Standard integral of 1/√(x²-a²)

A question is this type if and only if it requires direct application of the standard result that ∫1/√(x²-a²)dx = arcosh(x/a) + c or ln|x + √(x²-a²)| + c.

3 questions · Challenging +1.1

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CAIE Further Paper 2 2024 November Q1
4 marks Standard +0.8
Find the value of \(\int_6^7 \frac{1}{\sqrt{(x-5)^2-1}} \, dx\), giving your answer in the form \(\ln(a + \sqrt{b})\), where \(a\) and \(b\) are integers to be determined. [4]
WJEC Further Unit 4 2019 June Q12
14 marks Challenging +1.2
  1. Evaluate \(\int_3^4 \frac{1}{\sqrt{x^2 - 4}} \mathrm{d}x\), giving your answer correct to three decimal places. [3]
  2. Given that \(\int_1^2 \frac{k}{9 - x^2} \mathrm{d}x = \ln \frac{25}{4}\), find the value of \(k\). [5]
  3. Show that \(\int \frac{(\cosh x - \sinh x)^3}{\cosh^2 x + \sinh^2 x - \sinh 2x} \mathrm{d}x\) can be expressed as \(-e^{-x} + c\), where \(c\) is a constant. [6]
WJEC Further Unit 4 2022 June Q7
8 marks Challenging +1.2
  1. Express \(4x^2 + 10x - 24\) in the form \(a(x + b)^2 + c\), where \(a\), \(b\), \(c\) are constants whose values are to be found. [3]
  2. Hence evaluate the integral $$\int_3^5 \frac{6}{\sqrt{4x^2 + 10x - 24}} dx.$$ Give your answer correct to 3 decimal places. [5]