| Exam Board | Edexcel |
|---|---|
| Module | F3 (Further Pure Mathematics 3) |
| Year | 2024 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Algebraic manipulation before substitution |
| Difficulty | Standard +0.8 This is a Further Maths F3 question requiring completing the square, partial fraction decomposition of a rational function with a surd denominator, and integration using substitution. While it involves multiple steps and careful algebraic manipulation, each individual technique is standard for Further Maths students. The structured parts guide students through the solution, making it moderately challenging but not requiring novel insight. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points4.08h Integration: inverse trig/hyperbolic substitutions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(4x^2+4x+17 = 4\!\left(x+\frac{1}{2}\right)^2 - 1 + 17 = (2x+1)^2 + 16\) | B1 | Either \(p\) or \(q\) correct |
| \(p=1,\ q=16\) | B1 | Both correct values. Values may be embedded within \((2x+p)^2+q\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(A=8,\ B=4\) | B1 | Both correct values (accept if embedded) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\displaystyle\int \frac{1}{\sqrt{(2x+1)^2+16}}\,dx + \int \frac{8x+4}{\sqrt{4x^2+4x+17}}\,dx\) | M1 | For \(\ldots\text{arsinh}(f(x))\), \(f(x)\neq k\) or logarithmic equivalent |
| \(= \frac{1}{2}\text{arsinh}\!\left(\frac{2x+1}{4}\right) + 2(4x^2+4x+17)^{\frac{1}{2}}\) | M1, A1 | M1: For \(\ldots(4x^2+4x+17)^{\frac{1}{2}}\) or \(\ldots((2x+1)^2+16)^{\frac{1}{2}}\). A1: Fully correct integration |
| Substitutes limits; \(\frac{1}{2}\text{arsinh}\!\left(\frac{3}{4}\right) - \frac{1}{2}\text{arsinh}\!\left(\frac{5}{12}\right) + 2\sqrt{25} - 2\sqrt{\frac{169}{9}}\) | ddM1 | Substitutes and subtracts with limits. Results from separate integrals must be combined. Requires both previous M marks. |
| \(= \frac{4}{3} + \frac{1}{2}\ln\frac{4}{3}\) | A1 | Correct answer in correct form. Allow \(k=\frac{4}{3}\) if \(k+\frac{1}{2}\ln k\) is seen. Algebraic integration must be used. |
## Question 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $4x^2+4x+17 = 4\!\left(x+\frac{1}{2}\right)^2 - 1 + 17 = (2x+1)^2 + 16$ | B1 | Either $p$ or $q$ correct |
| $p=1,\ q=16$ | B1 | Both correct values. Values may be embedded within $(2x+p)^2+q$ |
## Question 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $A=8,\ B=4$ | B1 | Both correct values (accept if embedded) |
## Question 5(c):
$\displaystyle\int_{\frac{1}{3}}^{1} \frac{8x+5}{\sqrt{4x^2+4x+17}}\,dx$
| Answer | Mark | Guidance |
|--------|------|----------|
| $\displaystyle\int \frac{1}{\sqrt{(2x+1)^2+16}}\,dx + \int \frac{8x+4}{\sqrt{4x^2+4x+17}}\,dx$ | M1 | For $\ldots\text{arsinh}(f(x))$, $f(x)\neq k$ or logarithmic equivalent |
| $= \frac{1}{2}\text{arsinh}\!\left(\frac{2x+1}{4}\right) + 2(4x^2+4x+17)^{\frac{1}{2}}$ | M1, A1 | M1: For $\ldots(4x^2+4x+17)^{\frac{1}{2}}$ or $\ldots((2x+1)^2+16)^{\frac{1}{2}}$. A1: Fully correct integration |
| Substitutes limits; $\frac{1}{2}\text{arsinh}\!\left(\frac{3}{4}\right) - \frac{1}{2}\text{arsinh}\!\left(\frac{5}{12}\right) + 2\sqrt{25} - 2\sqrt{\frac{169}{9}}$ | ddM1 | Substitutes and subtracts with limits. Results from separate integrals must be combined. Requires both previous M marks. |
| $= \frac{4}{3} + \frac{1}{2}\ln\frac{4}{3}$ | A1 | Correct answer in correct form. Allow $k=\frac{4}{3}$ if $k+\frac{1}{2}\ln k$ is seen. Algebraic integration must be used. |5.
$$4 x ^ { 2 } + 4 x + 17 \equiv ( 2 x + p ) ^ { 2 } + q$$
where $p$ and $q$ are integers.
\begin{enumerate}[label=(\alph*)]
\item Determine the value of $p$ and the value of $q$
Given that
$$\frac { 8 x + 5 } { \sqrt { 4 x ^ { 2 } + 4 x + 17 } } \equiv \frac { 1 } { \sqrt { 4 x ^ { 2 } + 4 x + 17 } } + \frac { A x + B } { \sqrt { 4 x ^ { 2 } + 4 x + 17 } }$$
where $A$ and $B$ are integers,
\item write down the value of $A$ and the value of $B$
\item Hence use algebraic integration to show that
$$\int _ { \frac { 1 } { 3 } } ^ { 1 } \frac { 8 x + 5 } { \sqrt { 4 x ^ { 2 } + 4 x + 17 } } \mathrm {~d} x = k + \frac { 1 } { 2 } \ln k$$
where $k$ is a rational number to be determined.
\end{enumerate}
\hfill \mbox{\textit{Edexcel F3 2024 Q5 [8]}}