OCR Further Pure Core 1 2018 December — Question 9 7 marks

Exam BoardOCR
ModuleFurther Pure Core 1 (Further Pure Core 1)
Year2018
SessionDecember
Marks7
TopicIntegration using inverse trig and hyperbolic functions
TypeStandard integral of 1/√(x²+a²)
DifficultyStandard +0.8 This is a Further Maths question requiring completion of the square, recognition of the inverse hyperbolic/logarithmic form of ∫1/√(x²+a²)dx, and careful evaluation of limits with exact simplification to ln(p+q√2) form. While the technique is standard for FP1, the algebraic manipulation and exact form requirement elevate it above typical A-level integration questions.
Spec4.08h Integration: inverse trig/hyperbolic substitutions
9 In this question you must show detailed reasoning. Find \(\int _ { - 1 } ^ { 11 } \frac { 1 } { \sqrt { x ^ { 2 } + 6 x + 13 } } \mathrm {~d} x\) giving your answer in the form \(\ln ( p + q \sqrt { 2 } )\) where \(p\) and \(q\) are integers to be determined.
AnswerMarks Guidance
\(\int_{-1}^{11} \frac{1}{\sqrt{x^2+6x+13}} dx\)M1 Attempt to complete the square
\(x^2 + 6x + 13 = (x+3)^2 + 4\); \(\Rightarrow \int_{-1}^{11} \frac{1}{\sqrt{x^2+6x+13}} dx = \int_{-1}^{11} \frac{1}{\sqrt{(x+3)^2+4}} dx\)A1 In a suitable form to integrate
\(= \left[\ln\left((x+3) + \sqrt{(x+3)^2+4}\right)\right]_{-1}^{11}\)M1, A1 Use standard form; Integration
\(= \ln\left(14 + \sqrt{200}\right) - \ln\left(2 + \sqrt{8}\right)\)M1, A1 Use correct limits to obtain answer in lns
\(= \ln\left(\frac{7+5\sqrt{2}}{1+\sqrt{2}}\right) = \ln\left(3 + 2\sqrt{2}\right)\); i.e. \(p = 3, q = 2\)A1 
$\int_{-1}^{11} \frac{1}{\sqrt{x^2+6x+13}} dx$ | M1 | Attempt to complete the square

$x^2 + 6x + 13 = (x+3)^2 + 4$; $\Rightarrow \int_{-1}^{11} \frac{1}{\sqrt{x^2+6x+13}} dx = \int_{-1}^{11} \frac{1}{\sqrt{(x+3)^2+4}} dx$ | A1 | In a suitable form to integrate

$= \left[\ln\left((x+3) + \sqrt{(x+3)^2+4}\right)\right]_{-1}^{11}$ | M1, A1 | Use standard form; Integration

$= \ln\left(14 + \sqrt{200}\right) - \ln\left(2 + \sqrt{8}\right)$ | M1, A1 | Use correct limits to obtain answer in lns

$= \ln\left(\frac{7+5\sqrt{2}}{1+\sqrt{2}}\right) = \ln\left(3 + 2\sqrt{2}\right)$; i.e. $p = 3, q = 2$ | A1 |

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9 In this question you must show detailed reasoning.
Find $\int _ { - 1 } ^ { 11 } \frac { 1 } { \sqrt { x ^ { 2 } + 6 x + 13 } } \mathrm {~d} x$ giving your answer in the form $\ln ( p + q \sqrt { 2 } )$ where $p$ and $q$ are integers to be determined.

\hfill \mbox{\textit{OCR Further Pure Core 1 2018 Q9 [7]}}
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