A2 June 2023 Paper 1 Q2

EdexcelCurrent spec6 marksHyperbolic Functions Integration

(a) Write \(x^2 + 4x - 5\) in the form \((x + p)^2 + q\) where \(p\) and \(q\) are integers. (1)

(b) Hence use a standard integral from the formula book to find\[\int \frac{1}{\sqrt{x^2 + 4x - 5}}\,\mathrm{d}x\] (2)

(c) Determine the mean value of the function\[\mathrm{f}(x) = \frac{1}{\sqrt{x^2 + 4x - 5}} \qquad 3 \leqslant x \leqslant 13\]giving your answer in the form \(A\ln B\) where \(A\) and \(B\) are constants in simplest form. (3)

Scheme	Marks	AO
\(x^2 + 4x - 5 = (x + 2)^2 - 9\)	B1	1.1b
	(1)

Mark scheme (b)

Scheme	Marks	AO
\(\displaystyle\int \frac{1}{\sqrt{(x + p)^2 - q}}\,\mathrm{d}x = \operatorname{arcosh}\left(\frac{x + p}{\sqrt{q}}\right)(+c)\) or \(\ln\left(x + p + \sqrt{(x + p)^2 - q}\right)(+c)\)	M1	1.1a
\(= \operatorname{arcosh}\left(\dfrac{x + 2}{3}\right)\) or \(\ln\left(x + 2 + \sqrt{(x + 2)^2 - 9}\right)\) oe	A1	2.2a
	(2)

Notes

M1: Achieves a correct form for the integration for their \(p\) and \(q\) from part (a):

\(\operatorname{arcosh}\left(\dfrac{x + p}{\sqrt{q}}\right)(+c)\) or \(\ln\left(x + p + \sqrt{(x + p)^2 - q}\right)(+c)\) or e.g. \(\ln\left(\dfrac{x + p}{\sqrt{q}} + \sqrt{\left(\dfrac{x + p}{\sqrt{q}}\right)^2 - 1}\right)(+c)\)
where \(p \neq 0,\ q \neq 1\)

Allow \(\cosh^{-1}\) for arcosh
Allow attempts that use substitution following an attempt to complete the square but must be an appropriate substitution e.g. \(x + p = \sqrt{q}\cosh u\) leading to a correct form as above.

A1: Correct integration. The “\(+ c\)” is not required. Apply isw once a correct expression is seen.
Note that \(\ln\left(\dfrac{x + 2}{3} + \sqrt{\left(\dfrac{x + 2}{3}\right)^2 - 1}\right)(+c)\) is also correct

Mark scheme (c)

Scheme	Marks	AO
Mean \(= \dfrac{1}{13 - 3}\displaystyle\int_3^{13} \frac{1}{\sqrt{x^2 + 4x - 5}}\,\mathrm{d}x\)	B1	1.2
\(\dfrac{1}{10}\displaystyle\int_3^{13} \frac{1}{\sqrt{x^2 + 4x - 5}}\,\mathrm{d}x = \frac{1}{10}\left(\operatorname{arcosh}\left(\frac{15}{3}\right) - \operatorname{arcosh}\left(\frac{5}{3}\right)\right)\) or \(\dfrac{1}{10}\displaystyle\int_3^{13} \frac{1}{\sqrt{x^2 + 4x - 5}}\,\mathrm{d}x = \frac{1}{10}\left(\ln\left(15 + \sqrt{216}\right) - \ln\left(5 + \sqrt{16}\right)\right)\)	M1	1.1b
\(= \dfrac{1}{10}\ln\left(\dfrac{5 + 2\sqrt{6}}{3}\right)\) or \(\dfrac{1}{20}\ln\left(\dfrac{49 + 20\sqrt{6}}{9}\right)\)	A1	3.2a
	(3)
	(6 marks)

Notes

B1: Recalls the definition of a mean function accurately. \(\dfrac{1}{13 - 3}\displaystyle\int_3^{13} \frac{1}{\sqrt{x^2 + 4x - 5}}\,\mathrm{d}x\) seen or implied.
Note that the \(\dfrac{1}{13 - 3}\) may appear at the end. \(\dfrac{1}{13 - 3}\displaystyle\int_3^{13} \mathrm{f}(x)\,\mathrm{d}x\) is sufficient as f(\(x\)) is defined in the question. Also allow it to be implied by e.g. \(\dfrac{1}{10}\left[\mathrm{g}(x)\right]_3^{13}\) where g(\(x\)) is their integrated function.

M1: Applies the correct limits the right way round to whatever they think the answer to part (b) is.
This can be awarded if the \(\dfrac{1}{10}\) is present or not.

A1: Correct answer in correct form. Allow equivalents e.g. \(\dfrac{1}{10}\ln\left(\dfrac{5}{3} + \dfrac{2\sqrt{6}}{3}\right)\), \(\dfrac{1}{20}\ln\left(\dfrac{49}{9} + \dfrac{20\sqrt{6}}{9}\right)\)
And allow if the surd is not simplified e.g. \(\dfrac{1}{10}\ln\left(\dfrac{5 + \sqrt{24}}{3}\right)\), \(\dfrac{1}{20}\ln\left(\dfrac{49 + \sqrt{2400}}{9}\right)\)
Apply isw once a correct answer is seen.
The brackets must be present in forms such as \(\dfrac{1}{10}\ln\left(\dfrac{5}{3} + \dfrac{2\sqrt{6}}{3}\right)\), \(\dfrac{1}{20}\ln\left(\dfrac{49}{9} + \dfrac{20\sqrt{6}}{9}\right)\) but not in e.g. \(\dfrac{1}{10}\ln\dfrac{5 + \sqrt{24}}{3}\)
If extra values are offered then score A0