| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2021 |
| Session | November |
| Marks | 7 |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Use trig identity before indefinite integration |
| Difficulty | Moderate -0.3 This question tests standard integration techniques: (a)(i) is a routine double-angle identity verification, (a)(ii) applies it directly to integrate cos²(4x) using substitution, and (b) requires the standard trick of writing sin³x = sin x(1-cos²x). All are textbook exercises with no novel problem-solving required, making it slightly easier than average but not trivial due to the multi-step nature and need to recall specific techniques. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
\begin{enumerate}[label=(\alph*)]
\item
\begin{enumerate}[label=(\roman*)]
\item Show that $\cos^2 x \equiv \frac{1}{2} + \frac{1}{2}\cos 2x$ [1]
\item Hence find $\int 2\cos^2 4x \, dx$ [3]
\end{enumerate}
\item Find $\int \sin^3 x \, dx$ [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2021 Q9 [7]}}