| Exam Board | Edexcel |
|---|---|
| Module | F3 (Further Pure Mathematics 3) |
| Year | 2022 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration using inverse trig and hyperbolic functions |
| Type | Completing square then standard inverse trig |
| Difficulty | Standard +0.8 This is a Further Maths F3 question requiring completing the square for inverse trig integration and a trigonometric substitution with exact value calculation. While these are standard techniques at FM level, the multi-step nature, algebraic manipulation required, and need for exact answers (not calculator-dependent) place it moderately above average difficulty. |
| Spec | 1.08h Integration by substitution4.08h Integration: inverse trig/hyperbolic substitutions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(5 + 4x - x^2 = 9 - (x-2)^2\) | B1 | Correct completion of the square, any correct result |
| \(\int \frac{1}{\sqrt{5+4x-x^2}}\, dx = \int \frac{1}{\sqrt{9-(x-2)^2}}\, dx = \sin^{-1}\left(\frac{x-2}{3}\right)(+c)\) | M1A1 | M1: Obtains \(k\sin^{-1}f(x)\). A1: Correct integration (\(+c\) not needed) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x = 6 \Rightarrow \theta = \frac{\pi}{3}\); \(x = 2\sqrt{3} \Rightarrow \theta = \frac{\pi}{6}\) | B1 | Correct \(\theta\) limits in radians |
| \(\int \frac{18}{(x^2-9)^{\frac{3}{2}}}\, dx = \int \frac{18 \times 3\sec\theta\tan\theta}{(9\sec^2\theta - 9)^{\frac{3}{2}}}\, d\theta\) | M1 | For \(\int \frac{18}{\left((3\sec\theta)^2 - 9\right)^{\frac{3}{2}}} \times \left(\text{their } \frac{dx}{d\theta}\right)d\theta\) |
| \(= 2\int \frac{\cos\theta}{\sin^2\theta}\, d\theta\) or \(2\int \frac{\sec\theta}{\tan^2\theta}\, d\theta\) | A1 | Correct simplified integral |
| \(2\int \frac{\cos\theta}{\sin^2\theta}\, d\theta = 2\int \csc\theta\cot\theta\, d\theta = -2\csc\theta (+c)\) | M1 | Obtains \(k\csc\theta (+c)\) |
| \(\left[-2\csc\theta\right]_{\frac{\pi}{6}}^{\frac{\pi}{3}} = -2\csc\frac{\pi}{3} + 2\csc\frac{\pi}{6}\) | dM1 | Uses changed limits correctly. Depends on all previous method marks |
| \(= 4 - \frac{4}{3}\sqrt{3}\) | A1 | Cao. Allow these 2 marks if limits given in degrees |
# Question 2:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $5 + 4x - x^2 = 9 - (x-2)^2$ | B1 | Correct completion of the square, any correct result |
| $\int \frac{1}{\sqrt{5+4x-x^2}}\, dx = \int \frac{1}{\sqrt{9-(x-2)^2}}\, dx = \sin^{-1}\left(\frac{x-2}{3}\right)(+c)$ | M1A1 | M1: Obtains $k\sin^{-1}f(x)$. A1: Correct integration ($+c$ not needed) |
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x = 6 \Rightarrow \theta = \frac{\pi}{3}$; $x = 2\sqrt{3} \Rightarrow \theta = \frac{\pi}{6}$ | B1 | Correct $\theta$ limits in radians |
| $\int \frac{18}{(x^2-9)^{\frac{3}{2}}}\, dx = \int \frac{18 \times 3\sec\theta\tan\theta}{(9\sec^2\theta - 9)^{\frac{3}{2}}}\, d\theta$ | M1 | For $\int \frac{18}{\left((3\sec\theta)^2 - 9\right)^{\frac{3}{2}}} \times \left(\text{their } \frac{dx}{d\theta}\right)d\theta$ |
| $= 2\int \frac{\cos\theta}{\sin^2\theta}\, d\theta$ or $2\int \frac{\sec\theta}{\tan^2\theta}\, d\theta$ | A1 | Correct simplified integral |
| $2\int \frac{\cos\theta}{\sin^2\theta}\, d\theta = 2\int \csc\theta\cot\theta\, d\theta = -2\csc\theta (+c)$ | M1 | Obtains $k\csc\theta (+c)$ |
| $\left[-2\csc\theta\right]_{\frac{\pi}{6}}^{\frac{\pi}{3}} = -2\csc\frac{\pi}{3} + 2\csc\frac{\pi}{6}$ | dM1 | Uses changed limits correctly. Depends on all previous method marks |
| $= 4 - \frac{4}{3}\sqrt{3}$ | A1 | Cao. Allow these 2 marks if limits given in degrees |
---\begin{enumerate}
\item In this question you must show all stages of your working.
\end{enumerate}
Solutions relying entirely on calculator technology are not acceptable.\\
(i) Determine
$$\int \frac { 1 } { \sqrt { 5 + 4 x - x ^ { 2 } } } d x$$
(ii) Use the substitution $x = 3 \sec \theta$ to determine the exact value of
$$\int _ { 2 \sqrt { 3 } } ^ { 6 } \frac { 18 } { \left( x ^ { 2 } - 9 \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$$
Give your answer in the form $A + B \sqrt { 3 }$ where $A$ and $B$ are constants to be found.
\hfill \mbox{\textit{Edexcel F3 2022 Q2 [9]}}