A2 June 2019 Paper 1 Q7

7 The function \(\mathrm{sech}\,x\) is defined by \(\mathrm{sech}\,x = \dfrac{1}{\cosh x}\).

(a) Show that \(\mathrm{sech}\,x = \dfrac{2\mathrm{e}^x}{\mathrm{e}^{2x} + 1}\). [2]

(b) Using a suitable substitution, find \(\displaystyle\int \mathrm{sech}\,x\,\mathrm{d}x\). [4]

Mark scheme (a)

Scheme	Marks	AO
\(\cosh x = \dfrac{\mathrm{e}^x + \mathrm{e}^{-x}}{2} = \dfrac{\mathrm{e}^{2x} + 1}{2\mathrm{e}^x}\)	M1	1.1
\(\Rightarrow \mathrm{sech}\,x = \dfrac{2\mathrm{e}^x}{\mathrm{e}^{2x} + 1}\) AG	A1	2.1
	[2]

M1: Use of \(\cosh x\) in exponentials

Mark scheme (b)

Scheme	Marks	AO
\(u = \mathrm{e}^x \Rightarrow \mathrm{d}u = \mathrm{e}^x\mathrm{d}x\) \(\Rightarrow \mathrm{d}x = \dfrac{\mathrm{d}u}{u}\)	M1	3.1a
\(\Rightarrow \displaystyle\int \mathrm{sech}\,x\,\mathrm{d}x = \int\left(\frac{2\mathrm{e}^x}{\mathrm{e}^{2x} + 1}\right)\mathrm{d}x\) \(\displaystyle = \int \frac{2u}{u^2 + 1}.\frac{\mathrm{d}u}{u}\)	A1	1.1
\(= 2\tan^{-1}(u) + c\)	M1	3.1a
\(= 2\tan^{-1}(\mathrm{e}^x) + c\)	A1	1.1
	[4]

M1: Substitute and use (a)

A1: any form entirely in terms of \(u\). Allow absence of \(\mathrm{d}u\)

M1: Use standard form for integral and substitute back

A1: Must include \(c\)

Scheme	Marks
\(u = \sinh x \Rightarrow \mathrm{d}u = \cosh x\,\mathrm{d}x\)	M1
\(\Rightarrow \displaystyle\int \mathrm{sech}\,x\,\mathrm{d}x = \int \frac{1}{\cosh x}.\frac{\mathrm{d}u}{\cosh x} = \int \frac{\mathrm{d}u}{\cosh^2 x}\) \(\displaystyle = \int \frac{\mathrm{d}u}{1 + \sinh^2 x} = \int \frac{\mathrm{d}u}{1 + u^2}\)	A1
\(= \tan^{-1}u + c\)	M1
\(= \tan^{-1}(\sinh x) + c\)	A1
	[4]

Scheme	Marks
\(\displaystyle\int \mathrm{sech}\,x\,\mathrm{d}x = \int \frac{2\mathrm{e}^x}{\mathrm{e}^{2x} + 1}\,\mathrm{d}x\) Let \(\mathrm{e}^x = \tan u \Rightarrow \mathrm{e}^x\mathrm{d}x = \sec^2 u\,\mathrm{d}u \Rightarrow \mathrm{d}x = \dfrac{\sec^2 u}{\tan u}\mathrm{d}u\)	M1
\(\Rightarrow \displaystyle\int \mathrm{sech}\,x\,\mathrm{d}x = \int \frac{2\tan u}{\tan^2 u + 1}.\frac{\sec^2 u}{\tan u}\,\mathrm{d}u = 2\int \mathrm{d}u\)	A1
\(= 2u + c\)	M1
\(= 2\tan^{-1}(\mathrm{e}^x) + c\)	A1
	[4]

M1: Substitute

A1: In correct form

M1: Integrate and substitute back

A1: Must include \(c\)