Question 2 - A-Level Maths

WJEC Further Unit 4 Specimen — Question 2 6 marks

Exam Board	WJEC
Module	Further Unit 4 (Further Unit 4)
Session	Specimen
Marks	6
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Integration using inverse trig and hyperbolic functions
Type	Standard integral of 1/√(x²+a²)
Difficulty	Challenging +1.2 This is a standard Further Maths integration requiring completion of the square to get the form 1/√(a² + u²), then applying the inverse sinh or arctan substitution formula. While it requires knowledge of hyperbolic/inverse trig integrals beyond standard A-level, it's a routine application of a known technique with straightforward algebra, making it moderately above average difficulty.
Spec	1.06f Laws of logarithms: addition, subtraction, power rules 1.08h Integration by substitution

Evaluate the integral $$\int_0^1 \frac{dx}{\sqrt{2x^2 + 4x + 6}}.$$ [6]

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Attempting to complete the square:

Answer	Marks	Guidance
$\text{Integral} = \int_0^1 \frac{dx}{\sqrt{2(x+1)^2 + 4}}$	M1	AO3
$= \frac{1}{\sqrt{2}} \int_0^1 \frac{dx}{\sqrt{(x+1)^2 + 2}}$	A1	AO3
$= \frac{1}{\sqrt{2}} \left[\sinh^{-1}\left(\frac{x+1}{\sqrt{2}}\right)\right]_0^1$	A1	AO3
$= \frac{1}{\sqrt{2}}\left(\sinh^{-1}\left(\frac{2}{\sqrt{2}}\right) - \sinh^{-1}\left(\frac{1}{\sqrt{2}}\right)\right)$	A1	AO3
$= 0.345$ (0.344882...)	A1	AO3

Total: [6]

Guidance: Award M0 for unsupported working

Attempting to complete the square:

$\text{Integral} = \int_0^1 \frac{dx}{\sqrt{2(x+1)^2 + 4}}$ | M1 | AO3

$= \frac{1}{\sqrt{2}} \int_0^1 \frac{dx}{\sqrt{(x+1)^2 + 2}}$ | A1 | AO3

$= \frac{1}{\sqrt{2}} \left[\sinh^{-1}\left(\frac{x+1}{\sqrt{2}}\right)\right]_0^1$ | A1 | AO3

$= \frac{1}{\sqrt{2}}\left(\sinh^{-1}\left(\frac{2}{\sqrt{2}}\right) - \sinh^{-1}\left(\frac{1}{\sqrt{2}}\right)\right)$ | A1 | AO3

$= 0.345$ (0.344882...) | A1 | AO3

**Total: [6]**

**Guidance:** Award M0 for unsupported working

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Evaluate the integral
$$\int_0^1 \frac{dx}{\sqrt{2x^2 + 4x + 6}}.$$ [6]

\hfill \mbox{\textit{WJEC Further Unit 4  Q2 [6]}}

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Q1 7 Q2 6 Q3 5 Q4 9 Q5 8 Q6 7 Q7 10 Q8 10 Q9 14 Q10 11 Q11 17 Q12 16

Answer	Marks	Guidance
\(\text{Integral} = \int_0^1 \frac{dx}{\sqrt{2(x+1)^2 + 4}}\)	M1	AO3
\(= \frac{1}{\sqrt{2}} \int_0^1 \frac{dx}{\sqrt{(x+1)^2 + 2}}\)	A1	AO3
\(= \frac{1}{\sqrt{2}} \left[\sinh^{-1}\left(\frac{x+1}{\sqrt{2}}\right)\right]_0^1\)	A1	AO3
\(= \frac{1}{\sqrt{2}}\left(\sinh^{-1}\left(\frac{2}{\sqrt{2}}\right) - \sinh^{-1}\left(\frac{1}{\sqrt{2}}\right)\right)\)	A1	AO3
\(= 0.345\) (0.344882...)	A1	AO3