Edexcel FP3 Specimen — Question 4

Exam BoardEdexcel
ModuleFP3 (Further Pure Mathematics 3)
SessionSpecimen
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicIntegration using inverse trig and hyperbolic functions
TypeHyperbolic substitution to evaluate integral
DifficultyChallenging +1.8 This is a Further Maths FP3 question requiring hyperbolic substitution (x = 2sinh u) and knowledge of hyperbolic identities to integrate. While the technique is standard for FP3, it requires multiple steps including substitution, using cosh²u - sinh²u = 1, integration by parts or double angle formulas, and back-substitution. The 7 marks reflect substantial working, placing it well above average difficulty but still a bookwork-style question for students who have learned the technique.
Spec1.08h Integration by substitution
Find \(\int \sqrt{x^2 + 4} \, dx\). (Total 7 marks)
AnswerMarks Guidance
\(\sqrt{(x^2 + 4)} = \sqrt{(4\sinh^2 t + 4)^{\frac{1}{2}}} = 2\cosh t\)B1
\(dx = 2\cosh t \, dt\)M1A1
\(I = \int\sqrt{(x^2 + 4)} dx = 4\int\cosh^2 t \, dt\)M1A1
\(= 2\int(\cosh 2t + 1) dt\)
\(= \sinh 2t + 2t + c\)M1A1
\(= \frac{1}{2}x\sqrt{(x^2 + 4)} + 2\operatorname{arsinh}\left(\frac{x}{2}\right) + c\)M1A1ft (7) 
$\sqrt{(x^2 + 4)} = \sqrt{(4\sinh^2 t + 4)^{\frac{1}{2}}} = 2\cosh t$ | B1 |

$dx = 2\cosh t \, dt$ | M1A1 |

$I = \int\sqrt{(x^2 + 4)} dx = 4\int\cosh^2 t \, dt$ | M1A1 |

$= 2\int(\cosh 2t + 1) dt$ | |

$= \sinh 2t + 2t + c$ | M1A1 |

$= \frac{1}{2}x\sqrt{(x^2 + 4)} + 2\operatorname{arsinh}\left(\frac{x}{2}\right) + c$ | M1A1ft | (7)

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Find $\int \sqrt{x^2 + 4} \, dx$.

(Total 7 marks)

\hfill \mbox{\textit{Edexcel FP3  Q4}}
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