| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| Topic | Integration with Partial Fractions |
| Type | Improper integrals with discontinuity |
| Difficulty | Standard +0.3 This tests basic improper integral evaluation at a singularity (x=0) with straightforward power rule integration. Part (i) converges trivially to 6, part (ii) diverges due to the stronger singularity. The conceptual understanding required (recognizing why integrals are improper and checking convergence) is elementary for Further Maths students, making this easier than average A-level material. |
| Spec | 4.08c Improper integrals: infinite limits or discontinuous integrands |
\begin{enumerate}[label=(\alph*)]
\item For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:
\begin{enumerate}[label=(\roman*)]
\item $\int_0^9 \frac{1}{\sqrt{x}} dx$; [3 marks]
\item $\int_0^9 \frac{1}{x\sqrt{x}} dx$. [3 marks]
\end{enumerate}
\item Explain briefly why the integrals in part (a) are improper integrals. [1 mark]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2022 Q1 [7]}}