| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | February |
| Marks | 7 |
| Topic | Partial Fractions |
| Type | Improper fraction with quadratic factor – division then partial fractions and integrate |
| Difficulty | Moderate -0.3 This is a straightforward improper fraction question with clear scaffolding. Part (a) involves expanding and comparing coefficients (routine algebra), while part (b) requires standard integration of polynomial and logarithmic terms. The identity is given rather than requiring students to perform polynomial division, making it easier than typical improper fraction questions. The integration is mechanical once the partial fraction form is obtained. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
12.
Given that
$$4 x ^ { 3 } + 2 x ^ { 2 } + 17 x + 8 \equiv ( A x + B ) \left( x ^ { 2 } + 4 \right) + C x + D$$
\begin{enumerate}[label=(\alph*)]
\item find the values of the constants $A , B , C$ and $D$.\\
(3)
\item Hence find
$$\int _ { 1 } ^ { 4 } \frac { 4 x ^ { 3 } + 2 x ^ { 2 } + 17 x + 8 } { x ^ { 2 } + 4 } d x$$
giving your answer in the form $p + \ln q$, where $p$ and $q$ are integers.
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q12 [7]}}