A2 October 2021 Paper 1 Q4
4 In this question you must show detailed reasoning.
Determine the mean value of \(\dfrac{1}{1 + 4x^2}\) between \(x = -1\) and \(x = 1\). Give your answer to 3 significant figures. [4]
| Scheme | Marks | AO |
|---|---|---|
| DR \(\displaystyle \text{mean} = \frac{1}{1 - (-1)}\int_{-1}^{1} \frac{1}{1 + 4x^2}\,\mathrm{d}x\) | B1 | 1.1 |
| \(\displaystyle = \frac{1}{8}\int_{-1}^{1} \frac{1}{\frac{1}{4} + x^2}\,\mathrm{d}x\) | M1 | 1.1 |
| \(= \dfrac{1}{8}\left[2\arctan 2x\right]_{-1}^{1}\) | A1 | 1.1 |
| \(= \dfrac{1}{2}\arctan 2 \approx 0.554\) | A1 | 1.1 |
| [4] |
Notes
M1: or \(\displaystyle u = 2x \Rightarrow \frac{1}{2}\int_{-2}^{2} \frac{1}{1 + u^2}\,\frac{1}{2}\,\mathrm{d}u\)
M1 for rearranging denominator correctly into appropriate form or for \(k\arctan 2x\)
A1: or \(= \dfrac{1}{4}\left[\arctan u\right]_{-2}^{2}\)