| Exam Board | Edexcel |
|---|---|
| Module | CP2 (Core Pure 2) |
| Year | 2019 |
| Session | June |
| 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 | Standard +0.8 This is a Further Maths Core Pure question requiring knowledge of hyperbolic function integration (a specialist topic), a substitution to manipulate the integral into standard form, and then application to find a mean value involving inverse hyperbolic functions expressed as logarithms. While methodical, it demands multiple advanced techniques beyond standard A-level and careful algebraic manipulation, placing it moderately above average difficulty. |
| Spec | 4.07e Inverse hyperbolic: definitions, domains, ranges4.08e Mean value of function: using integral4.08h Integration: inverse trig/hyperbolic substitutions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x = \frac{3}{2}\sinh u\) | B1 | Selects appropriate substitution leading to integrable form |
| \(\int \frac{dx}{\sqrt{4x^2+9}} = \int \frac{1}{\sqrt{4(\frac{9}{4})\sinh^2 u+9}} \times \frac{3}{2}\cosh u \, du\) | M1 | Fully correct method for substitution including substituting into function and dealing with \(dx\) |
| \(= \int \frac{1}{2} \, du\) | A1 | Correct simplified integral in terms of \(u\) |
| \(= \frac{1}{2}u = \frac{1}{2}\sinh^{-1}\left(\frac{2x}{3}\right) + c\) | A1 | Correct answer including \(+c\); allow arcsinh or arsinh for \(\sinh^{-1}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x = \frac{3}{2}\tan u\) | B1 | Selects appropriate substitution |
| \(\int \frac{dx}{\sqrt{4x^2+9}} = \int \frac{1}{\sqrt{4(\frac{9}{4})\tan^2 u+9}} \times \frac{3}{2}\sec^2 u \, du\) | M1 | Fully correct method for substitution |
| \(= \int \frac{1}{2}\sec u \, du\) | A1 | Correct simplified integral |
| \(= \frac{1}{2}\ln(\sec u + \tan u) = \frac{1}{2}\ln\left(\frac{2x}{3}+\sqrt{1+\left(\frac{2x}{3}\right)^2}\right) = \frac{1}{2}\sinh^{-1}\left(\frac{2x}{3}\right)+c\) | A1 | Correct answer including \(+c\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x = \frac{1}{2}u\) or \(x = ku\) where \(k>0\), \(k\neq 1\) | B1 | Selects appropriate substitution |
| \(\int \frac{dx}{\sqrt{4x^2+9}} = \int \frac{1}{\sqrt{4(\frac{1}{4})u^2+9}} \times \frac{1}{2} \, du\) | M1 | Fully correct method |
| \(= \frac{1}{2}\int \frac{1}{\sqrt{u^2+9}} \, du\) | A1 | Correct simplified integral |
| \(= \frac{1}{2}\sinh^{-1}\frac{u}{3} = \frac{1}{2}\sinh^{-1}\frac{2x}{3}+c\) | A1 | Correct answer including \(+c\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Mean value \(= \frac{1}{3-0}\left[\frac{1}{2}\sinh^{-1}\left(\frac{2x}{3}\right)\right]_0^3 = \frac{1}{3}\times\frac{1}{2}\sinh^{-1}\left(\frac{2\times3}{3}\right)-(-0)\) | M1 | Correctly applies mean value method; substitutes limits 0 and 3; condone omission of 0 |
| \(= \frac{1}{6}\ln(2+\sqrt{5})\) (Brackets required) | A1ft | Correct exact answer (follow through on \(A\) and \(B\)); brackets required if appropriate |
# Question 3(a) Way 1:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x = \frac{3}{2}\sinh u$ | B1 | Selects appropriate substitution leading to integrable form |
| $\int \frac{dx}{\sqrt{4x^2+9}} = \int \frac{1}{\sqrt{4(\frac{9}{4})\sinh^2 u+9}} \times \frac{3}{2}\cosh u \, du$ | M1 | Fully correct method for substitution including substituting into function and dealing with $dx$ |
| $= \int \frac{1}{2} \, du$ | A1 | Correct simplified integral in terms of $u$ |
| $= \frac{1}{2}u = \frac{1}{2}\sinh^{-1}\left(\frac{2x}{3}\right) + c$ | A1 | Correct answer including $+c$; allow arcsinh or arsinh for $\sinh^{-1}$ |
# Question 3(a) Way 2:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x = \frac{3}{2}\tan u$ | B1 | Selects appropriate substitution |
| $\int \frac{dx}{\sqrt{4x^2+9}} = \int \frac{1}{\sqrt{4(\frac{9}{4})\tan^2 u+9}} \times \frac{3}{2}\sec^2 u \, du$ | M1 | Fully correct method for substitution |
| $= \int \frac{1}{2}\sec u \, du$ | A1 | Correct simplified integral |
| $= \frac{1}{2}\ln(\sec u + \tan u) = \frac{1}{2}\ln\left(\frac{2x}{3}+\sqrt{1+\left(\frac{2x}{3}\right)^2}\right) = \frac{1}{2}\sinh^{-1}\left(\frac{2x}{3}\right)+c$ | A1 | Correct answer including $+c$ |
# Question 3(a) Way 3:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x = \frac{1}{2}u$ or $x = ku$ where $k>0$, $k\neq 1$ | B1 | Selects appropriate substitution |
| $\int \frac{dx}{\sqrt{4x^2+9}} = \int \frac{1}{\sqrt{4(\frac{1}{4})u^2+9}} \times \frac{1}{2} \, du$ | M1 | Fully correct method |
| $= \frac{1}{2}\int \frac{1}{\sqrt{u^2+9}} \, du$ | A1 | Correct simplified integral |
| $= \frac{1}{2}\sinh^{-1}\frac{u}{3} = \frac{1}{2}\sinh^{-1}\frac{2x}{3}+c$ | A1 | Correct answer including $+c$ |
# Question 3(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Mean value $= \frac{1}{3-0}\left[\frac{1}{2}\sinh^{-1}\left(\frac{2x}{3}\right)\right]_0^3 = \frac{1}{3}\times\frac{1}{2}\sinh^{-1}\left(\frac{2\times3}{3}\right)-(-0)$ | M1 | Correctly applies mean value method; substitutes limits 0 and 3; condone omission of 0 |
| $= \frac{1}{6}\ln(2+\sqrt{5})$ **(Brackets required)** | A1ft | Correct exact answer (follow through on $A$ and $B$); brackets required if appropriate |
---3.
$$f ( x ) = \frac { 1 } { \sqrt { 4 x ^ { 2 } + 9 } }$$
\begin{enumerate}[label=(\alph*)]
\item Using a substitution, that should be stated clearly, show that
$$\int \mathrm { f } ( x ) \mathrm { d } x = A \sinh ^ { - 1 } ( B x ) + c$$
where $c$ is an arbitrary constant and $A$ and $B$ are constants to be found.
\item Hence find, in exact form in terms of natural logarithms, the mean value of $\mathrm { f } ( x )$ over the interval $[ 0,3 ]$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel CP2 2019 Q3 [6]}}