| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | June |
| Marks | 8 |
| Topic | Integration by Substitution |
| Type | Show definite integral equals specific value (trigonometric substitution or identity) |
| Difficulty | Challenging +1.2 This is a Further Maths integration question requiring trigonometric manipulation and careful analysis of discontinuities. Part (i) involves splitting the integrand, using substitution or standard results, and simplifying logarithms—moderately challenging but follows established techniques. Part (ii) tests understanding of when integrals are undefined due to discontinuities in the domain, which is conceptually important but straightforward to identify. The 'detailed reasoning' requirement and the need to manipulate the final answer into a specific form elevate this above routine, but it remains a standard FM Pure integration exercise without requiring novel insight. |
| Spec | 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
In this question you must use detailed reasoning.
\begin{enumerate}[label=(\roman*)]
\item Show that $\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1+\sin 2x}{-\cos 2x} dx = \ln(\sqrt{a} + b)$, where $a$ and $b$ are integers to be determined. [6]
\item Show that $\int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} \frac{1+\sin 2x}{-\cos 2x} dx$ is undefined, explaining your reasoning clearly. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q15 [8]}}