October 2021 Paper 3 Q7
7 Determine \(\displaystyle\int x\cos 2x\,\mathrm{d}x\). [3]
| Scheme | Marks | AO |
|---|---|---|
| Let \(u = x\), \(\dfrac{\mathrm{d}v}{\mathrm{d}x} = \cos 2x\) \(v = \frac{1}{2}\sin 2x\) \(\displaystyle\int x\cos 2x\,\mathrm{d}x = \frac{1}{2}x\sin 2x - \int \frac{1}{2}\sin 2x\,\mathrm{d}x\) | M1 M1 | 3.1a 1.1 |
| \(\frac{1}{2}x\sin 2x + \frac{1}{4}\cos 2x + c\) | A1 | 2.5 |
| [3] |
Notes
M1: Parts with \(u = x\), \(\dfrac{\mathrm{d}v}{\mathrm{d}x} = \cos 2x\)
M1: Allow if \(-\left[\frac{1}{2}x\sin 2x - \int\frac{1}{2}\sin 2x\,\mathrm{d}x\right]\) or if 1 error
A1: \(+c\) needed for A1