| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2008 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Derive triple angle then evaluate integral |
| Difficulty | Standard +0.3 Part (a) is a standard derivation using addition and double angle formulae with clear guidance. Part (b) requires rearranging the identity and integrating, which is routine once the connection is made. This is a typical C4 question testing formula manipulation and integration technique, slightly above average due to the two-part structure and need to connect the parts, but well within standard A-level scope. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int \sin^3 x \, dx = \int \sin^2 x \sin x \, dx\) | M1 | identify parts and attempt to integrate |
| Answer | Marks | Guidance |
|---|---|---|
| \(= -\sin^2 x \cos x - \frac{2}{3}\cos^3 x \quad (+C)\) | A2 | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int \sin^3 x \, dx = \int \sin^2 x \, d(-\cos x)\) | M1 | condone sign error |
| Answer | Marks | Guidance |
|---|---|---|
| \(= -\cos x + \frac{1}{3}\cos^3 x \quad (+C)\) | A2 | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int \sin x(1-\cos^2 x) \, dx\) | M1 | this form and attempt to integrate |
| \(= -\cos x + \frac{1}{3}\cos^3 x \quad (+C)\) | A2 | 3 |
## Question 3(b):
**Alternative Method 1:**
$\int \sin^3 x \, dx = \int \sin^2 x \sin x \, dx$ | M1 | identify parts and attempt to integrate
$= -\sin^2 x \cos x - \int -2\cos x \sin x \cos x \, dx$
$= -\sin^2 x \cos x - \frac{2}{3}\cos^3 x \quad (+C)$ | A2 | 3
---
**Alternative Method 2:**
$\int \sin^3 x \, dx = \int \sin^2 x \, d(-\cos x)$ | M1 | condone sign error
$= \int -(1-\cos^2 x) \, d(\cos x)$
$= -\cos x + \frac{1}{3}\cos^3 x \quad (+C)$ | A2 | 3
---
**Alternative Method 3:**
$\int \sin x \sin^2 x \, dx$
$\int \sin x(1-\cos^2 x) \, dx$ | M1 | this form and attempt to integrate
$= -\cos x + \frac{1}{3}\cos^3 x \quad (+C)$ | A2 | 3
---3
\begin{enumerate}[label=(\alph*)]
\item By writing $\sin 3 x$ as $\sin ( x + 2 x )$, show that $\sin 3 x = 3 \sin x - 4 \sin ^ { 3 } x$ for all values of $x$.
\item Hence, or otherwise, find $\int \sin ^ { 3 } x \mathrm {~d} x$.
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2008 Q3 [8]}}