Standard integral of 1/√(a²-x²)

A question is this type if and only if it requires direct application of the standard result that ∫1/√(a²-x²)dx = arcsin(x/a) + c, possibly after completing the square or substitution.

6 questions · Standard +0.5

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CAIE Further Paper 2 2020 Specimen Q2
6 marks Standard +0.8
2 Fid \(\mathbf { b }\) ex ct le \(\mathbf { 6 } \int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 3 + 4 x - 4 x ^ { 2 } } } \mathrm {~d} x\).
Edexcel FP3 2016 June Q4
12 marks Standard +0.8
4. (i) Find, without using a calculator, $$\int _ { 3 } ^ { 5 } \frac { 1 } { \sqrt { 15 + 2 x - x ^ { 2 } } } d x$$ giving your answer as a multiple of \(\pi\).
(ii)
  1. Show that $$5 \cosh x - 4 \sinh x = \frac { \mathrm { e } ^ { 2 x } + 9 } { 2 \mathrm { e } ^ { x } }$$
  2. Hence, using the substitution \(u = e ^ { x }\) or otherwise, find $$\int \frac { 1 } { 5 \cosh x - 4 \sinh x } d x$$
OCR FP2 2015 June Q3
5 marks Standard +0.3
3 By first completing the square, find the exact value of \(\int _ { \frac { 1 } { 2 } } ^ { 1 } \frac { 1 } { \sqrt { 2 x - x ^ { 2 } } } \mathrm {~d} x\).
AQA FP2 2012 January Q6
8 marks Standard +0.3
6
  1. Express \(7 + 4 x - 2 x ^ { 2 }\) in the form \(a - b ( x - c ) ^ { 2 }\), where \(a , b\) and \(c\) are integers.
  2. By means of a suitable substitution, or otherwise, find the exact value of $$\int _ { 1 } ^ { \frac { 5 } { 2 } } \frac { \mathrm {~d} x } { \sqrt { 7 + 4 x - 2 x ^ { 2 } } }$$
OCR MEI Further Pure Core 2020 November Q3
4 marks Standard +0.3
3 In this question you must show detailed reasoning.
Find \(\int _ { 0 } ^ { \frac { 1 } { 3 } } \frac { 1 } { \sqrt { 4 - 9 x ^ { 2 } } } \mathrm {~d} x\), expressing your answer in terms of \(\pi\).
OCR MEI Further Pure Core 2023 June Q15
5 marks Standard +0.3
15 In this question you must show detailed reasoning. Evaluate \(\int _ { 1 } ^ { 2 } \frac { 1 } { \sqrt { 1 + 2 x - x ^ { 2 } } } d x\), giving your answer in terms of \(\pi\).