| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2023 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Improper algebraic form then partial fractions |
| Difficulty | Standard +0.3 This is a standard improper fraction requiring polynomial long division followed by straightforward integration. The algebraic manipulation is routine for P3 level, and the integration of each term (including the partial fraction remainder) uses basic formulas. It's slightly easier than average due to the denominator being a perfect square, avoiding the need to split into multiple partial fractions. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Any correct constant: \(A=2\) or \(B=3\) or \(C=-1\) or \(D=5\) | B1 | One correct constant or one correct term in \(Ax^2 + Bx + C + \frac{D}{(x+3)^2}\) |
| \(2x^4+15x^3+35x^2+21x-4 = Ax^2(x+3)^2+Bx(x+3)^2+C(x+3)^2+D\) \(\Rightarrow A=\ldots, B=\ldots, C=\ldots, D=\ldots\) or via long division | M1 | Complete method for finding \(A\), \(B\), \(C\) and \(D\). Via substitution/comparing coefficients minimum required is identity of correct form (condoning slips) followed by values for \(A\), \(B\), \(C\), \(D\). Via division: look for divisor of \(x^2+6x+9\), quotient that is quadratic and remainder that is linear or constant |
| 2 correct of \(A=2, B=3, C=-1, D=5\) | A1 | 2 correct constants following award of M1. Note: first division gives \(2x^3+9x^2+8x-3\) with remainder of 5 |
| \(A=2, B=3, C=-1, D=5\) | A1 | All correct following award of M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\int f(x)\,dx = \int\left(2x^2+3x-1+\frac{5}{(x+3)^2}\right)dx = \frac{2x^3}{3}+\frac{3x^2}{2}-x-\frac{5}{x+3}(+c)\) | M1 A1ft A1 | M1: \(\int \frac{D}{(x+3)^2}\,dx \to \frac{k}{x+3}\) where \(k\) is constant. May be awarded following a term \(\int\frac{\alpha x+D}{(x+3)^2}\,dx\) following division — look for this being correctly split. A1ft: Follow through on their \(D\). A1: Fully correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int\left(Ax^2 + Bx + C + \frac{D}{(x+3)^2}\right)dx = \frac{Ax^3}{3} + \frac{Bx^2}{2} + Cx - \frac{D}{x+3}\) | A1ft | Correct integration following through on their non-zero constants. Allow with \(A\), \(B\), \(C\), \(D\) as above or with made up values |
| All correct with or without \(+c\) | A1 | Allow \(-1x\) for \(-x\) |
# Question 4:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| Any correct constant: $A=2$ or $B=3$ or $C=-1$ or $D=5$ | B1 | One correct constant or one correct term in $Ax^2 + Bx + C + \frac{D}{(x+3)^2}$ |
| $2x^4+15x^3+35x^2+21x-4 = Ax^2(x+3)^2+Bx(x+3)^2+C(x+3)^2+D$ $\Rightarrow A=\ldots, B=\ldots, C=\ldots, D=\ldots$ or via long division | M1 | Complete method for finding $A$, $B$, $C$ and $D$. Via substitution/comparing coefficients minimum required is identity of correct form (condoning slips) followed by values for $A$, $B$, $C$, $D$. Via division: look for divisor of $x^2+6x+9$, quotient that is quadratic and remainder that is linear or constant |
| 2 correct of $A=2, B=3, C=-1, D=5$ | A1 | 2 correct constants following award of M1. Note: first division gives $2x^3+9x^2+8x-3$ with remainder of 5 |
| $A=2, B=3, C=-1, D=5$ | A1 | All correct following award of M1 |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $\int f(x)\,dx = \int\left(2x^2+3x-1+\frac{5}{(x+3)^2}\right)dx = \frac{2x^3}{3}+\frac{3x^2}{2}-x-\frac{5}{x+3}(+c)$ | M1 A1ft A1 | M1: $\int \frac{D}{(x+3)^2}\,dx \to \frac{k}{x+3}$ where $k$ is constant. May be awarded following a term $\int\frac{\alpha x+D}{(x+3)^2}\,dx$ following division — look for this being correctly split. A1ft: Follow through on their $D$. A1: Fully correct |
## Question A1ft:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int\left(Ax^2 + Bx + C + \frac{D}{(x+3)^2}\right)dx = \frac{Ax^3}{3} + \frac{Bx^2}{2} + Cx - \frac{D}{x+3}$ | A1ft | Correct integration following through on their non-zero constants. Allow with $A$, $B$, $C$, $D$ as above or with made up values |
| All correct with or without $+c$ | A1 | Allow $-1x$ for $-x$ |
---4.
$$f ( x ) = \frac { 2 x ^ { 4 } + 15 x ^ { 3 } + 35 x ^ { 2 } + 21 x - 4 } { ( x + 3 ) ^ { 2 } } \quad x \in \mathbb { R } \quad x > - 3$$
\begin{enumerate}[label=(\alph*)]
\item Find the values of the constants $A$, $B$, $C$ and $D$ such that
$$\mathrm { f } ( x ) = A x ^ { 2 } + B x + C + \frac { D } { ( x + 3 ) ^ { 2 } }$$
\item Hence find,
$$\int \mathrm { f } ( x ) \mathrm { d } x$$
\end{enumerate}
\hfill \mbox{\textit{Edexcel P3 2023 Q4 [7]}}