| Exam Board | Edexcel |
|---|---|
| Module | F3 (Further Pure Mathematics 3) |
| Year | 2023 |
| Session | January |
| Marks | 5 |
| 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 question requiring knowledge of standard inverse hyperbolic integrals (∫1/√(x²+a²) = arcsinh(x/a) + C), followed by evaluation of definite integral with exact form manipulation. While the pattern is standard for FM students, the algebraic manipulation to reach the specific logarithmic form with surds requires careful execution and is above average difficulty even for Further Maths. |
| Spec | 4.08h Integration: inverse trig/hyperbolic substitutions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int \frac{1}{\sqrt{9x^2+16}}\,dx = \frac{1}{3}\int \frac{1}{\sqrt{x^2+\frac{16}{9}}}\,dx\) \(= \frac{1}{3}\,\text{arsinh}\!\left(\frac{3x}{4}\right)\) or \(\frac{1}{3}\,\text{arsinh}\!\left(\frac{x}{\frac{4}{3}}\right)\) \((+c)\) or \(\frac{1}{3}\ln\!\left(x+\sqrt{x^2+\left(\frac{4}{3}\right)^2}\right)(+c)\) | M1 A1 | M1: Obtains \(p\,\text{arsinh}(qx)\) or \(r\ln\!\left\{x+\sqrt{x^2+s}\right\}\) or \(t\ln(ux+\sqrt{vx^2+w})\), \(p,q,r,s,t,u,v,w>0\); A1: Any correct expression, may be unsimplified; "+c" not required; allow \(\sinh^{-1}\); "arcsin"/"arsin" is M0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\left[\frac{1}{3}\,\text{arsinh}\!\left(\frac{3x}{4}\right)\right]_{-2}^{2}\) or \(\left[\frac{1}{3}\ln\!\left(x+\sqrt{x^2+\frac{16}{9}}\right)\right]_{-2}^{2}\) | M1 | Substitutes limits 2 and \(-2\) into expression of form \(p\,\text{arsinh}(qx)\) or \(r\ln\!\left\{x+\sqrt{x^2+s}\right\}\) or \(t\ln(ux+\sqrt{vx^2+w})\) and subtracts; no rounded decimals unless exact values recovered |
| \(\frac{1}{3}\ln\!\left(\frac{11}{2}+\frac{3\sqrt{13}}{2}\right)\) or \(\frac{1}{3}\ln\frac{11+3\sqrt{13}}{2}\) or \(\frac{2}{3}\ln\!\left(\frac{3}{2}+\frac{\sqrt{13}}{2}\right)\) or \(\frac{2}{3}\ln\frac{3+\sqrt{13}}{2}\) | dM1 A1 | dM1: Obtains form \(a\ln(b+c\sqrt{13})\) or \(a\ln\!\left(\frac{d+e\sqrt{13}}{f}\right)\) where \(a,b,c,d,e,f\) exact and \(>0\); requires previous M mark; A1: Any correct equivalent, correct bracketing if needed; must come from correct work; allow \(a=\frac{2}{3}, b=\frac{3}{2}, c=\frac{1}{2}\) |
# Question 4(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int \frac{1}{\sqrt{9x^2+16}}\,dx = \frac{1}{3}\int \frac{1}{\sqrt{x^2+\frac{16}{9}}}\,dx$ $= \frac{1}{3}\,\text{arsinh}\!\left(\frac{3x}{4}\right)$ or $\frac{1}{3}\,\text{arsinh}\!\left(\frac{x}{\frac{4}{3}}\right)$ $(+c)$ or $\frac{1}{3}\ln\!\left(x+\sqrt{x^2+\left(\frac{4}{3}\right)^2}\right)(+c)$ | M1 A1 | M1: Obtains $p\,\text{arsinh}(qx)$ or $r\ln\!\left\{x+\sqrt{x^2+s}\right\}$ or $t\ln(ux+\sqrt{vx^2+w})$, $p,q,r,s,t,u,v,w>0$; A1: Any correct expression, may be unsimplified; "+c" not required; allow $\sinh^{-1}$; "arcsin"/"arsin" is M0 |
**(2)**
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# Question 4(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\left[\frac{1}{3}\,\text{arsinh}\!\left(\frac{3x}{4}\right)\right]_{-2}^{2}$ or $\left[\frac{1}{3}\ln\!\left(x+\sqrt{x^2+\frac{16}{9}}\right)\right]_{-2}^{2}$ | M1 | Substitutes limits 2 and $-2$ into expression of form $p\,\text{arsinh}(qx)$ or $r\ln\!\left\{x+\sqrt{x^2+s}\right\}$ or $t\ln(ux+\sqrt{vx^2+w})$ and subtracts; no rounded decimals unless exact values recovered |
| $\frac{1}{3}\ln\!\left(\frac{11}{2}+\frac{3\sqrt{13}}{2}\right)$ or $\frac{1}{3}\ln\frac{11+3\sqrt{13}}{2}$ or $\frac{2}{3}\ln\!\left(\frac{3}{2}+\frac{\sqrt{13}}{2}\right)$ or $\frac{2}{3}\ln\frac{3+\sqrt{13}}{2}$ | dM1 A1 | dM1: Obtains form $a\ln(b+c\sqrt{13})$ or $a\ln\!\left(\frac{d+e\sqrt{13}}{f}\right)$ where $a,b,c,d,e,f$ exact and $>0$; requires previous M mark; A1: Any correct equivalent, correct bracketing if needed; must come from correct work; allow $a=\frac{2}{3}, b=\frac{3}{2}, c=\frac{1}{2}$ |
**(3) Total: 5**
---\begin{enumerate}
\item (a) Determine
\end{enumerate}
$$\int \frac { 1 } { \sqrt { 9 x ^ { 2 } + 16 } } \mathrm {~d} x$$
(b) Hence determine the exact value of
$$\int _ { - 2 } ^ { 2 } \frac { 1 } { \sqrt { 9 x ^ { 2 } + 16 } } d x$$
Give your answer in the form $a \ln ( b + c \sqrt { 13 } )$, where $a , b$ and $c$ are rational numbers.
\hfill \mbox{\textit{Edexcel F3 2023 Q4 [5]}}