| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | October |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Algebraic manipulation before substitution |
| Difficulty | Standard +0.3 This is a standard P3 integration question requiring polynomial long division followed by routine substitution. The algebraic manipulation in part (a) is methodical but straightforward, and part (b) uses the standard result ∫f'(x)/f(x)dx = ln|f(x)|. While multi-step, it follows a well-practiced template with no novel insight required, making it slightly easier than average. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08d Evaluate definite integrals: between limits1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(2x^3 - 4x - 15 = (Ax+B)(x^2+3x+4) + C(2x+3) \Rightarrow A = ...\) or long division getting term \(2x...\) | M1 | Correct method leading to at least one constant; long division or comparing coefficients or substituting values |
| \(A = 2\) | A1 | May be seen in long division or in expression |
| \(B = -6, C = 3\) | A1A1 | Either \(B=-6\) or \(C=3\); both \(B=-6\) and \(C=3\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int f(x)\,dx = "A"\frac{x^2}{2} + "B"x + "C"\ln(x^2+3x+4)\) | M1 A1ft | Look for \(x^n \to x^{n+1}\) at least once; \(\frac{2x+3}{x^2+3x+4} \to k\ln(x^2+3x+4)\); correct integration following their \(A\), \(B\), \(C\) |
| \(\int f(x)\,dx = x^2 - 6x + 3\ln(x^2+3x+4)\) | ||
| \(\int_3^5 f(x)\,dx = \bigl(5^2-6\times5+3\ln(5^2+3\times5+4)\bigr) - \bigl(3^2-6\times3+3\ln(3^2+3\times3+4)\bigr) = ..\) | dM1 | Applies limits and subtracts; must have numerical values |
| \(= -5+3\ln44+9-3\ln22 = 4+3\ln\!\left(\frac{44}{22}\right)\) | M1 | Uses correct log laws to combine log terms into single log term |
| \(= 4 + \ln 8\) | A1 | Correct answer in the form specified |
# Question 1:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $2x^3 - 4x - 15 = (Ax+B)(x^2+3x+4) + C(2x+3) \Rightarrow A = ...$ or long division getting term $2x...$ | M1 | Correct method leading to at least one constant; long division or comparing coefficients or substituting values |
| $A = 2$ | A1 | May be seen in long division or in expression |
| $B = -6, C = 3$ | A1A1 | Either $B=-6$ or $C=3$; both $B=-6$ and $C=3$ |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int f(x)\,dx = "A"\frac{x^2}{2} + "B"x + "C"\ln(x^2+3x+4)$ | M1 A1ft | Look for $x^n \to x^{n+1}$ at least once; $\frac{2x+3}{x^2+3x+4} \to k\ln(x^2+3x+4)$; correct integration following their $A$, $B$, $C$ |
| $\int f(x)\,dx = x^2 - 6x + 3\ln(x^2+3x+4)$ | | |
| $\int_3^5 f(x)\,dx = \bigl(5^2-6\times5+3\ln(5^2+3\times5+4)\bigr) - \bigl(3^2-6\times3+3\ln(3^2+3\times3+4)\bigr) = ..$ | dM1 | Applies limits and subtracts; must have numerical values |
| $= -5+3\ln44+9-3\ln22 = 4+3\ln\!\left(\frac{44}{22}\right)$ | M1 | Uses correct log laws to combine log terms into single log term |
| $= 4 + \ln 8$ | A1 | Correct answer in the form specified |
---\begin{enumerate}
\item In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
\end{enumerate}
$$f ( x ) = \frac { 2 x ^ { 3 } - 4 x - 15 } { x ^ { 2 } + 3 x + 4 }$$
(a) Show that
$$f ( x ) \equiv A x + B + \frac { C ( 2 x + 3 ) } { x ^ { 2 } + 3 x + 4 }$$
where $A , B$ and $C$ are integers to be found.\\
(b) Hence, find
$$\int _ { 3 } ^ { 5 } \mathrm { f } ( x ) \mathrm { d } x$$
giving your answer in the form $p + \ln q$, where $p$ and $q$ are integers.\\
\hfill \mbox{\textit{Edexcel P3 2022 Q1 [9]}}