| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2011 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Basic integration by parts |
| Difficulty | Moderate -0.3 This is a straightforward single application of integration by parts with standard functions (polynomial × trig). The limits are clean (0 and π/2) making evaluation simple, and it's a direct textbook-style question requiring only one technique with no additional complications or insight needed. Slightly easier than average due to its directness. |
| Spec | 1.08i Integration by parts |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(\int x\sin 2x\,dx = -\frac{x\cos 2x}{2} + \int \frac{\cos 2x}{2}\,dx\) | M1 A1 A1 | Integration by parts, correct first term, correct integral remaining |
| \(= \ldots + \frac{\sin 2x}{4}\) | M1 | Integration of \(\cos 2x\) term |
| \(\left[\ldots\right]_0^{\frac{\pi}{2}} = \frac{\pi}{4}\) | M1 A1 | Applying limits correctly, correct final answer |
| Total | [6] |
# Question 1:
| Working/Answer | Marks | Guidance |
|---|---|---|
| $\int x\sin 2x\,dx = -\frac{x\cos 2x}{2} + \int \frac{\cos 2x}{2}\,dx$ | M1 A1 A1 | Integration by parts, correct first term, correct integral remaining |
| $= \ldots + \frac{\sin 2x}{4}$ | M1 | Integration of $\cos 2x$ term |
| $\left[\ldots\right]_0^{\frac{\pi}{2}} = \frac{\pi}{4}$ | M1 A1 | Applying limits correctly, correct final answer |
| **Total** | **[6]** | |
---\begin{enumerate}
\item Use integration to find the exact value of
\end{enumerate}
$$\int _ { 0 } ^ { \frac { \pi } { 2 } } x \sin 2 x \mathrm {~d} x$$
\hfill \mbox{\textit{Edexcel C4 2011 Q1 [6]}}