| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2009 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Integrate using hyperbolic substitution |
| Difficulty | Standard +0.8 This is an FP2 hyperbolic substitution question requiring students to identify x = cosh θ, transform the integral, then integrate cosh² θ using the identity cosh 2θ = 2cosh² θ - 1, and finally back-substitute. While the technique is standard for FP2, it requires multiple steps including hyperbolic identities, integration, and careful algebraic manipulation. The question is moderately challenging but follows a predictable pattern for this module, placing it somewhat above average difficulty. |
| Spec | 4.07f Inverse hyperbolic: logarithmic forms4.08h Integration: inverse trig/hyperbolic substitutions |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Let \(x = \cosh \theta\) such that \(dx = \sinh \theta \, d\theta\). Clearly use \(\cosh^2\theta - \sinh^2\theta = 1\) | M1, A1, B1 | Clearly derive A.G. |
| (ii) Replace \(\cosh^2\theta\). Attempt to integrate their expression. Get \(\frac{1}{4}\sinh 2\theta + \frac{1}{2}\theta\) (+c) | M1, M1, A1 | Allow \(a(\cosh 2\theta \pm 1)\); Allow \(b\sinh 2\theta \pm a\theta\); Condone no +c |
| Clearly replace for \(x\) to A.G. | B1 | SC Use expo. def"; three terms. Attempt to integrate. Get \(\frac{1}{3}(e^{2\theta}-e^{-2\theta}) + \frac{1}{2}\theta\) (+c). Clearly replace for \(x\) to A.G. |
(i) Let $x = \cosh \theta$ such that $dx = \sinh \theta \, d\theta$. Clearly use $\cosh^2\theta - \sinh^2\theta = 1$ | M1, A1, B1 | Clearly derive A.G.
(ii) Replace $\cosh^2\theta$. Attempt to integrate their expression. Get $\frac{1}{4}\sinh 2\theta + \frac{1}{2}\theta$ (+c) | M1, M1, A1 | Allow $a(\cosh 2\theta \pm 1)$; Allow $b\sinh 2\theta \pm a\theta$; Condone no +c
Clearly replace for $x$ to A.G. | B1 | SC Use expo. def"; three terms. Attempt to integrate. Get $\frac{1}{3}(e^{2\theta}-e^{-2\theta}) + \frac{1}{2}\theta$ (+c). Clearly replace for $x$ to A.G. | M1, M1, A1, B1\begin{enumerate}[label=(\roman*)]
\item By means of a suitable substitution, show that
$$\int \frac{x^2}{\sqrt{x^2-1}} dx$$
can be transformed to $\int \cosh^2 \theta \, d\theta$. [2]
\item Hence show that $\int \frac{x^2}{\sqrt{x^2-1}} dx = \frac{1}{2}\sqrt{x^2-1} + \frac{1}{2}\cosh^{-1} x + c$. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR FP2 2009 Q4 [6]}}