| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2010 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Multi-part with preliminary simplification |
| Difficulty | Standard +0.3 Part (a) is routine trigonometric identity manipulation using double angle formulas. Part (b) requires integration by parts with the simplified form, which is a standard C4 technique with straightforward execution. The question is slightly above average due to the two-part structure and need to apply integration by parts, but both components are textbook-standard with no novel insight required. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.08i Integration by parts |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(f(\theta) = 4\cos^2\theta - 3\sin^2\theta = 4\left(\frac{1}{2} + \frac{1}{2}\cos 2\theta\right) - 3\left(\frac{1}{2} - \frac{1}{2}\cos 2\theta\right)\) | M1 M1 |
| \(= \frac{1}{2} + \frac{1}{2}\cos 2\theta\) ⭐ | cso | |
| A1 | ||
| (b) | \(\int \theta \cos 2\theta d\theta = \frac{1}{2}\theta \sin 2\theta - \frac{1}{2}\int \sin 2\theta d\theta = \frac{1}{2}\theta \sin 2\theta + \frac{1}{4}\cos 2\theta\) | M1 A1 A1 |
| \(\int \theta f(\theta) d\theta = \frac{1}{4}\theta^2 + \frac{7}{4}\theta \sin 2\theta + \frac{7}{8}\cos 2\theta\) | M1 A1 | |
| \(\left[\ldots\right]_0^{\pi/2} = \left[\frac{\pi^2}{16} + 0 - \frac{7}{8}\right] - \left[0 + 0 + \frac{7}{8}\right] = \frac{\pi^2}{16} - \frac{7}{4}\) | M1 A1 |
(a) | $f(\theta) = 4\cos^2\theta - 3\sin^2\theta = 4\left(\frac{1}{2} + \frac{1}{2}\cos 2\theta\right) - 3\left(\frac{1}{2} - \frac{1}{2}\cos 2\theta\right)$ | M1 M1 | |
| $= \frac{1}{2} + \frac{1}{2}\cos 2\theta$ ⭐ | | cso |
| | A1 | |
(b) | $\int \theta \cos 2\theta d\theta = \frac{1}{2}\theta \sin 2\theta - \frac{1}{2}\int \sin 2\theta d\theta = \frac{1}{2}\theta \sin 2\theta + \frac{1}{4}\cos 2\theta$ | M1 A1 A1 | |
| $\int \theta f(\theta) d\theta = \frac{1}{4}\theta^2 + \frac{7}{4}\theta \sin 2\theta + \frac{7}{8}\cos 2\theta$ | M1 A1 | |
| $\left[\ldots\right]_0^{\pi/2} = \left[\frac{\pi^2}{16} + 0 - \frac{7}{8}\right] - \left[0 + 0 + \frac{7}{8}\right] = \frac{\pi^2}{16} - \frac{7}{4}$ | M1 A1 | |
---6.
$$f ( \theta ) = 4 \cos ^ { 2 } \theta - 3 \sin ^ { 2 } \theta$$
\begin{enumerate}[label=(\alph*)]
\item Show that $f ( \theta ) = \frac { 1 } { 2 } + \frac { 7 } { 2 } \cos 2 \theta$.
\item Hence, using calculus, find the exact value of $\int _ { 0 } ^ { \frac { \pi } { 2 } } \theta \mathrm { f } ( \theta ) \mathrm { d } \theta$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 2010 Q6 [10]}}