Completing square then standard inverse trig

A question is this type if and only if it requires completing the square in a quadratic expression and then applying a standard inverse trigonometric or hyperbolic integral formula.

11 questions · Standard +0.6

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CAIE Further Paper 2 2024 June Q1
5 marks Standard +0.8
1 Find the exact value of \(\int _ { 2 } ^ { \frac { 7 } { 2 } } \frac { 1 } { \sqrt { 4 x - x ^ { 2 } - 1 } } \mathrm {~d} x\).
Edexcel F3 2022 January Q5
7 marks Standard +0.8
5. Determine
  1. \(\int \frac { 1 } { \sqrt { x ^ { 2 } - 3 x + 5 } } \mathrm {~d} x\)
  2. \(\int \frac { 1 } { \sqrt { 63 + 4 x - 4 x ^ { 2 } } } \mathrm {~d} x\)
Edexcel F3 2016 June Q3
12 marks Standard +0.8
3. Without using a calculator, find
  1. \(\int _ { - 2 } ^ { 1 } \frac { 1 } { x ^ { 2 } + 4 x + 13 } \mathrm {~d} x\), giving your answer as a multiple of \(\pi\),
  2. \(\int _ { - 1 } ^ { 4 } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 12 x + 34 } } \mathrm {~d} x\), giving your answer in the form \(p \ln ( q + r \sqrt { 2 } )\),
    where \(p , q\) and \(r\) are rational numbers to be found.
Edexcel F3 2020 June Q2
8 marks Standard +0.3
2. Determine
  1. \(\int \frac { 1 } { 3 x ^ { 2 } + 12 x + 24 } \mathrm {~d} x\)
  2. \(\int \frac { 1 } { \sqrt { 27 - 6 x - x ^ { 2 } } } \mathrm {~d} x\)
Edexcel F3 2022 June Q2
9 marks Standard +0.8
  1. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
  1. Determine $$\int \frac { 1 } { \sqrt { 5 + 4 x - x ^ { 2 } } } d x$$
  2. Use the substitution \(x = 3 \sec \theta\) to determine the exact value of $$\int _ { 2 \sqrt { 3 } } ^ { 6 } \frac { 18 } { \left( x ^ { 2 } - 9 \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$$ Give your answer in the form \(A + B \sqrt { 3 }\) where \(A\) and \(B\) are constants to be found.
Edexcel F3 2018 Specimen Q3
12 marks Standard +0.8
3. Without using a calculator, find
  1. \(\int _ { - 2 } ^ { 1 } \frac { 1 } { x ^ { 2 } + 4 x + 13 } \mathrm {~d} x\), giving your answer as a multiple of \(\pi\),
  2. \(\int _ { - 1 } ^ { 4 } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 12 x + 34 } } \mathrm {~d} x\), giving your answer in the form \(p \ln ( q + r \sqrt { 2 } )\),
    where \(p , q\) and \(r\) are rational numbers to be found.
    VIIIV SIHI NI J14M 10N OCVIIN SIHI NI III HM ION OOVERV SIHI NI JIIIM ION OO
Edexcel FP3 2011 June Q3
9 marks Standard +0.8
  1. Show that
    1. \(\int _ { 5 } ^ { 8 } \frac { 1 } { x ^ { 2 } - 10 x + 34 } \mathrm {~d} x = k \pi\), giving the value of the fraction \(k\),
    2. \(\int _ { 5 } ^ { 8 } \frac { 1 } { \sqrt { } \left( x ^ { 2 } - 10 x + 34 \right) } \mathrm { d } x = \ln ( A + \sqrt { } n )\), giving the values of the integers \(A\) and \(n\).
    $$I _ { n } = \int _ { 1 } ^ { \mathrm { e } } x ^ { 2 } ( \ln x ) ^ { n } \mathrm {~d} x , \quad n \geqslant 0$$
  2. Prove that, for \(n \geqslant 1\), $$I _ { n } = \frac { \mathrm { e } ^ { 3 } } { 3 } - \frac { n } { 3 } I _ { n - 1 }$$
  3. Find the exact value of \(I _ { 3 }\).
Edexcel FP3 2014 June Q2
7 marks Standard +0.3
2. $$9 x ^ { 2 } + 6 x + 5 \equiv a ( x + b ) ^ { 2 } + c$$
  1. Find the values of the constants \(a\), \(b\) and \(c\). Hence, or otherwise, find
  2. \(\int \frac { 1 } { 9 x ^ { 2 } + 6 x + 5 } d x\)
  3. \(\int \frac { 1 } { \sqrt { 9 x ^ { 2 } + 6 x + 5 } } \mathrm {~d} x\)
Edexcel FP3 2014 June Q3
8 marks Standard +0.8
  1. Using calculus, find the exact value of
    1. \(\int _ { 1 } ^ { 2 } \frac { 1 } { \sqrt { \left( x ^ { 2 } - 2 x + 3 \right) } } \mathrm { d } x\)
    2. \(\int _ { 0 } ^ { 1 } \mathrm { e } ^ { 2 x } \sinh x \mathrm {~d} x\)
WJEC Further Unit 4 2022 June Q7
8 marks Standard +0.3
  1. (a) Express \(4 x ^ { 2 } + 10 x - 24\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b , c\) are constants whose values are to be found.
    (b) Hence evaluate the integral
$$\int _ { 3 } ^ { 5 } \frac { 6 } { \sqrt { 4 x ^ { 2 } + 10 x - 24 } } \mathrm {~d} x$$ Give your answer correct to 3 decimal places.
Edexcel CP1 2023 June Q2
6 marks Standard +0.3
  1. (a) Write \(x ^ { 2 } + 4 x - 5\) in the form \(( x + p ) ^ { 2 } + q\) where \(p\) and \(q\) are integers.
    (b) Hence use a standard integral from the formula book to find
$$\int \frac { 1 } { \sqrt { x ^ { 2 } + 4 x - 5 } } \mathrm {~d} x$$ (c) Determine the mean value of the function $$\mathrm { f } ( x ) = \frac { 1 } { \sqrt { x ^ { 2 } + 4 x - 5 } } \quad 3 \leqslant x \leqslant 13$$ giving your answer in the form \(A \ln B\) where \(A\) and \(B\) are constants in simplest form.