| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2021 |
| Session | November |
| Marks | 6 |
| Topic | Partial Fractions |
| Type | Repeated linear factor with distinct linear factor – decompose and integrate |
| Difficulty | Moderate -0.3 Part (a) is a standard partial fractions decomposition with a repeated linear factor—routine A-level technique. Part (b) is straightforward integration using the given decomposition, requiring only basic logarithm and power rule integration. Both parts are textbook exercises with no problem-solving insight required, making this slightly easier than average. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
\begin{enumerate}[label=(\alph*)]
\item Express $\frac{5x+7}{(x+3)(x+1)^2}$ in partial fractions.
In this question you must show all of your algebraic steps clearly. [3]
The function $f(x) = \frac{2-6x+5x^2}{x^2(1-2x)}$ can be written in the form;
$$f(x) = \frac{-2}{x} + \frac{2}{x^2} + \frac{1}{1-2x}$$
\item Hence find the exact value of $\int_2^3 \frac{2-6x+5x^2}{x^2(1-2x)} dx$ [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2021 Q2 [6]}}