| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Algebraic manipulation before substitution |
| Difficulty | Moderate -0.3 This is a straightforward integration by substitution where the numerator is essentially the derivative of the denominator. Part (a) requires recognizing the standard form ∫f'(x)/f(x)dx = ln|f(x)| + c, which is a common pattern. Part (b) involves simple algebraic manipulation of logarithms to find k. While it requires showing working and understanding the technique, it's a routine application with no conceptual challenges, making it slightly easier than average. |
| Spec | 1.08d Evaluate definite integrals: between limits1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int \frac{9x}{3x^2+k}\,dx = \frac{3}{2}\ln(3x^2+k)+C\) | M1A1 | M1 integrates to \(A\ln(3x^2+k)\), allow \(A=1\), condone invisible brackets; A1 must include constant of integration different from \(k\), must have brackets around \(3x^2+k\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{3}{2}\ln(75+k) - \frac{3}{2}\ln(12+k) = \ln 8\) | M1 | Applies limits 5 and 2 to their integral |
| \(\frac{3}{2}\ln\!\left(\frac{75+k}{12+k}\right) = \ln 8\) | dM1 | Combines logarithms correctly |
| \(\frac{75+k}{12+k} = 4 \Rightarrow k = \ldots\) | ddM1 | Correct process to solve for \(k\) |
| \(k = 9\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Substitutes \(x=5\) and \(x=2\) into expression containing both \(x\) and \(k\), subtracting either way and setting equal to \(\ln 8\) | M1 | Condone slips if attempting to evaluate when substituting. Ignore the \(+C\) |
| Applies subtraction (or addition) law of logarithms: \(\pm A\ln(B+k) \pm A\ln(C+k) = \ln D\), e.g. \(A\ln\left(\frac{B+k}{C+k}\right) = \ln D\) | dM1 | Allow \(A=1\); condone arithmetic errors and invisible brackets |
| Solves equation of form \(\ln\left(\frac{B+k}{C+k}\right) = \ln D^{\frac{1}{A}}\) or \(\ln\left(\frac{B+k}{C+k}\right)^A = \ln D\), where \(A \neq 1\), removing logs, forming linear equation in \(k\), finding \(k\) | ddM1 | e.g. \(\ln\left(\frac{75+k}{12+k}\right) = \ln 8^{\frac{2}{3}} \Rightarrow 75+k = 48+4k\); condone arithmetic slips; condone if \(k\) is negative |
| \(k = 9\) cao | A1 | Intermediate working must be seen (e.g. equation with lns removed). Cannot proceed directly from \(\frac{3}{2}\ln(75+k)-\frac{3}{2}\ln(12+k)=\ln 8\) to \(k=9\) without intermediate working |
## Question 3:
### Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int \frac{9x}{3x^2+k}\,dx = \frac{3}{2}\ln(3x^2+k)+C$ | M1A1 | M1 integrates to $A\ln(3x^2+k)$, allow $A=1$, condone invisible brackets; A1 must include constant of integration different from $k$, must have brackets around $3x^2+k$ |
### Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{3}{2}\ln(75+k) - \frac{3}{2}\ln(12+k) = \ln 8$ | M1 | Applies limits 5 and 2 to their integral |
| $\frac{3}{2}\ln\!\left(\frac{75+k}{12+k}\right) = \ln 8$ | dM1 | Combines logarithms correctly |
| $\frac{75+k}{12+k} = 4 \Rightarrow k = \ldots$ | ddM1 | Correct process to solve for $k$ |
| $k = 9$ | A1 | |
# Question (b) [Logarithms question]:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Substitutes $x=5$ and $x=2$ into expression containing both $x$ and $k$, subtracting either way and setting equal to $\ln 8$ | M1 | Condone slips if attempting to evaluate when substituting. Ignore the $+C$ |
| Applies subtraction (or addition) law of logarithms: $\pm A\ln(B+k) \pm A\ln(C+k) = \ln D$, e.g. $A\ln\left(\frac{B+k}{C+k}\right) = \ln D$ | dM1 | Allow $A=1$; condone arithmetic errors and invisible brackets |
| Solves equation of form $\ln\left(\frac{B+k}{C+k}\right) = \ln D^{\frac{1}{A}}$ or $\ln\left(\frac{B+k}{C+k}\right)^A = \ln D$, where $A \neq 1$, removing logs, forming linear equation in $k$, finding $k$ | ddM1 | e.g. $\ln\left(\frac{75+k}{12+k}\right) = \ln 8^{\frac{2}{3}} \Rightarrow 75+k = 48+4k$; condone arithmetic slips; condone if $k$ is negative |
| $k = 9$ cao | A1 | Intermediate working must be seen (e.g. equation with lns removed). Cannot proceed directly from $\frac{3}{2}\ln(75+k)-\frac{3}{2}\ln(12+k)=\ln 8$ to $k=9$ without intermediate working |
---3. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.\\
Given that $k$ is a positive constant,
\begin{enumerate}[label=(\alph*)]
\item find
$$\int \frac { 9 x } { 3 x ^ { 2 } + k } d x$$
Given also that
$$\int _ { 2 } ^ { 5 } \frac { 9 x } { 3 x ^ { 2 } + k } \mathrm {~d} x = \ln 8$$
\item find the value of $k$
\end{enumerate}
\hfill \mbox{\textit{Edexcel P3 2022 Q3 [6]}}