A2 June 2019 Paper 1 Q13

OCR MEICurrent spec11 marksHyperbolic Functions Integration

(a) Using the logarithmic form of \(\operatorname{arcosh} x\), prove that the derivative of \(\operatorname{arcosh} x\) is \(\dfrac{1}{\sqrt{x^2 - 1}}\). [5]

(b) Hence find \(\displaystyle\int_1^2 \operatorname{arcosh} x\,\mathrm{d}x\), giving your answer in exact logarithmic form. [5]

(c) Ali tries to evaluate \(\displaystyle\int_0^1 \operatorname{arcosh} x\,\mathrm{d}x\) using his calculator, and gets an ‘error’. Explain why. [1]

Mark scheme (a)

Scheme	Marks	AO
\(y = \operatorname{arcosh} x = \ln\left(x + \sqrt{x^2 - 1}\right)\)	M1
	M1	1.1b
\(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{1 + x(x^2 - 1)^{-1/2}}{\left(x + (x^2 - 1)^{1/2}\right)}\)	A1	1.1b
\(= \dfrac{(x^2 - 1)^{1/2} + x}{\left(x + (x^2 - 1)^{1/2}\right)(x^2 - 1)^{1/2}}\)	A1	2.1
\(= \dfrac{1}{(x^2 - 1)^{1/2}}\,*\)	A1cao	2.2a
	[5]

Notes

M1: \(\dfrac{\mathrm{d}}{\mathrm{d}u}(\ln u) = \dfrac{1}{u}\)

M1: chain rule on \(\sqrt{x^2 - 1}\)

A1: correct expression

A1: or \(\dfrac{\left[(x^2 - 1)^{1/2} + x\right](x^2 - 1)^{-1/2}}{\left(x + (x^2 - 1)^{1/2}\right)}\)

A1cao: NB AG

Or \(\mathrm{e}^y = x + \sqrt{x^2 - 1}\); \(\mathrm{e}^y\,\mathrm{d}y/\mathrm{d}x = \ldots\) M1 \(= 1 + x(x^2 - 1)^{-1/2}\) M1 [substituting for \(\mathrm{e}^y\)]

Mark scheme (b)

Scheme	Marks	AO
let \(u = \operatorname{arcosh} x\), \(u' = 1/\sqrt{x^2 - 1}\), \(v' = 1\), \(v = x\)	M1	3.1a
\(\displaystyle\int_1^2 \operatorname{arcosh} x\,\mathrm{d}x = \big[x\operatorname{arcosh} x\big]_1^2 - \int_1^2 \frac{x}{\sqrt{x^2 - 1}}\,\mathrm{d}x\)	A1	1.1b
	M1	1.1b
\(= \left[x\operatorname{arcosh} x - \sqrt{x^2 - 1}\right]\)	A1	1.1b
\(= 2\operatorname{arcosh} 2 - \operatorname{arcosh} 1 - \sqrt{3}\) \(= 2\ln(2 + \sqrt{3}) - \sqrt{3}\)	A1cao
	[5]

Notes

M1: integration by parts; ignore limits

M1: substitution or inspection: \(\displaystyle\int \frac{x}{\sqrt{x^2 - 1}}\,\mathrm{d}x = \sqrt{x^2 - 1}\)

A1cao: oe e.g. \(\ln(7 + 4\sqrt{3}) - \sqrt{3}\); isw, not ln 1

Alternative solution

Scheme	Marks
Let \(x = \cosh u\), \(\mathrm{d}x = \sinh u\,\mathrm{d}u\) \(\displaystyle\int \operatorname{arcosh} x\,\mathrm{d}x = \int u\sinh u\,\mathrm{d}u\)	M1
\(\displaystyle = \big[u\cosh u\big] - \int \cosh u\,\mathrm{d}u\)	M1 A1
\(= \big[u\cosh u - \sinh u\big]_{\operatorname{arcosh} 1}^{\operatorname{arcosh} 2}\)	A1
\(= 2\ln(2 + \sqrt{3}) - \sqrt{3}\)	A1cao
	[5]

M1 A1: integration by parts

A1: limits not needed

A1cao: oe e.g. \(\ln(7 + 4\sqrt{3}) - \sqrt{3}\) [isw], not ln 1

Mark scheme (c)

Scheme	Marks	AO
\(\operatorname{arcosh} x\) does not exist for \(x \lt 1\)	B1	2.4
	[1]

Notes

B1: or \(\sqrt{x^2 - 1} = \sqrt{-1}\) not real, so \(\ln\left(x + \sqrt{x^2 - 1}\right)\) is not real; accept other valid arguments