| Exam Board | Edexcel |
|---|---|
| Module | F3 (Further Pure Mathematics 3) |
| Year | 2022 |
| Session | January |
| Marks | 7 |
| 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 question requiring completion of the square followed by recognition and application of standard inverse hyperbolic/trig integral forms. While the technique is systematic once learned, it requires multiple steps (completing the square, substitution, recognizing the standard form) and is beyond standard A-level. The second part with a negative quadratic adds mild complexity. Moderately challenging for Further Maths students but a standard exercise in the F3 syllabus. |
| Spec | 4.08g Derivatives: inverse trig and hyperbolic functions4.08h Integration: inverse trig/hyperbolic substitutions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x^2 - 3x + 5 = \left(x - \frac{3}{2}\right)^2 + \frac{11}{4}\) | B1 | Correct completion of the square |
| \(\int \frac{1}{\sqrt{x^2 - 3x + 5}}\, dx = \int \frac{1}{\sqrt{\left(x-\frac{3}{2}\right)^2 + \frac{11}{4}}}\, dx = \sinh^{-1}\frac{2x-3}{\sqrt{11}} (+c)\) | M1A1 | M1: Use of \(\sinh^{-1}\). A1: Fully correct expression (condone omission of \(+c\)). Allow equivalent correct expressions e.g. \(\sinh^{-1}\frac{x-\frac{3}{2}}{\frac{\sqrt{11}}{4}}(+c)\), \(\sinh^{-1}\frac{x-\frac{3}{2}}{\frac{\sqrt{11}}{2}}(+c)\). Allow equivalents for \(\sinh^{-1}\) e.g. arsinh, arcsinh but not arsin or arcsin. Logarithmic forms also accepted. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(63 + 4x - 4x^2 = -4\left(x^2 - x - \frac{63}{4}\right) = -4\left(\left(x - \frac{1}{2}\right)^2 - \frac{64}{4}\right)\) giving \(-4\left(\left(x-\frac{1}{2}\right)^2 - 16\right)\) or \(64 - 4\left(x - \frac{1}{2}\right)^2\) or \(64 - (2x-1)^2\) | M1 | Obtains \(-4\left(\left(x-\frac{1}{2}\right)^2 \pm \ldots\right)\) or \(-4\left(x-\frac{1}{2}\right)^2 \pm \ldots\) or \(\ldots-(2x-1)^2\) |
| \(64 - 4\left(x - \frac{1}{2}\right)^2\) or \(64-(2x-1)^2\) | A1 | Correct completion of the square |
| \(\int \frac{1}{\sqrt{63 + 4x - 4x^2}}\, dx = \frac{1}{2}\sin^{-1}\left(\frac{2x-1}{8}\right)(+c)\) | M1A1 | M1: Use of \(\sin^{-1}\). A1: Fully correct expression (condone omission of \(+c\)). Allow equivalent correct expressions e.g. \(\frac{1}{2}\sin^{-1}\frac{x-\frac{1}{2}}{4}(+c)\), \(-\frac{1}{2}\sin^{-1}\frac{\frac{1}{2}-x}{4}(+c)\). Allow equivalents for \(\sin^{-1}\) e.g. arsin, arcsin but not arsinh or arcsinh. |
## Question 5:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x^2 - 3x + 5 = \left(x - \frac{3}{2}\right)^2 + \frac{11}{4}$ | B1 | Correct completion of the square |
| $\int \frac{1}{\sqrt{x^2 - 3x + 5}}\, dx = \int \frac{1}{\sqrt{\left(x-\frac{3}{2}\right)^2 + \frac{11}{4}}}\, dx = \sinh^{-1}\frac{2x-3}{\sqrt{11}} (+c)$ | M1A1 | M1: Use of $\sinh^{-1}$. A1: Fully correct expression (condone omission of $+c$). Allow equivalent correct expressions e.g. $\sinh^{-1}\frac{x-\frac{3}{2}}{\frac{\sqrt{11}}{4}}(+c)$, $\sinh^{-1}\frac{x-\frac{3}{2}}{\frac{\sqrt{11}}{2}}(+c)$. Allow equivalents for $\sinh^{-1}$ e.g. arsinh, arcsinh but **not** arsin or arcsin. Logarithmic forms also accepted. |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $63 + 4x - 4x^2 = -4\left(x^2 - x - \frac{63}{4}\right) = -4\left(\left(x - \frac{1}{2}\right)^2 - \frac{64}{4}\right)$ giving $-4\left(\left(x-\frac{1}{2}\right)^2 - 16\right)$ or $64 - 4\left(x - \frac{1}{2}\right)^2$ or $64 - (2x-1)^2$ | M1 | Obtains $-4\left(\left(x-\frac{1}{2}\right)^2 \pm \ldots\right)$ or $-4\left(x-\frac{1}{2}\right)^2 \pm \ldots$ or $\ldots-(2x-1)^2$ |
| $64 - 4\left(x - \frac{1}{2}\right)^2$ or $64-(2x-1)^2$ | A1 | Correct completion of the square |
| $\int \frac{1}{\sqrt{63 + 4x - 4x^2}}\, dx = \frac{1}{2}\sin^{-1}\left(\frac{2x-1}{8}\right)(+c)$ | M1A1 | M1: Use of $\sin^{-1}$. A1: Fully correct expression (condone omission of $+c$). Allow equivalent correct expressions e.g. $\frac{1}{2}\sin^{-1}\frac{x-\frac{1}{2}}{4}(+c)$, $-\frac{1}{2}\sin^{-1}\frac{\frac{1}{2}-x}{4}(+c)$. Allow equivalents for $\sin^{-1}$ e.g. arsin, arcsin but **not** arsinh or arcsinh. |
---5. Determine\\
(i) $\int \frac { 1 } { \sqrt { x ^ { 2 } - 3 x + 5 } } \mathrm {~d} x$\\
(ii) $\int \frac { 1 } { \sqrt { 63 + 4 x - 4 x ^ { 2 } } } \mathrm {~d} x$
\hfill \mbox{\textit{Edexcel F3 2022 Q5 [7]}}