June 2023 Paper 2 Q10
10 Determine the exact value of \(\displaystyle\int_0^{\frac{\pi}{4}} 4x\cos 2x\,\mathrm{d}x\). [5]
| Scheme | Marks | AO |
|---|---|---|
| \(4x \times \frac{1}{2}\sin 2x - \int \frac{1}{2}\sin 2x \times 4\,\mathrm{d}x\) | M1* | 3.1a |
| \(2x\sin 2x - \int 2\sin 2x\,\mathrm{d}x\) oe | A1 | 1.1 |
| \(\mathrm{F}[x] = 2x\sin 2x + \cos 2x\) | A1 | 1.1 |
| \(\mathrm{F}\left[\frac{\pi}{4}\right] - \mathrm{F}[0]\) | M1dep* | 1.1 |
| \(\frac{\pi}{2} - 1\) or \(\frac{\pi - 2}{2}\) isw | A1 | 3.2a |
| [5] |
Notes
M1*: allow sign errors only, condone omission of \(\mathrm{d}x\)
A1: allow omission of \(\mathrm{d}x\); may be unsimplified
A1: ignore \(+\,c\)
M1dep*: must see substitution if \(\mathrm{F}[x]\) incorrect, otherwise may be implied by correct answer;
A1: allow recovery from bracket error