A2 June 2022 Paper 1 Q2
2 In this question you must show detailed reasoning.
Find the exact value of \(\displaystyle\int_3^{\infty} \frac{1}{x^2 - 4x + 5}\,\mathrm{d}x\). [5]
| Scheme | Marks | AO |
|---|---|---|
| DR \(\displaystyle\int_3^k \frac{1}{x^2 - 4x + 5}\,\mathrm{d}x = \int_3^k \frac{1}{(x - 2)^2 + 1}\,\mathrm{d}x\) | M1 | 3.1a |
| \(= \left[\arctan(x - 2)\right]_3^k\) | A1 | 1.1 |
| \(\displaystyle\lim_{k \to \infty}\left[\arctan(k - 2)\right] = \frac{\pi}{2}\) | E1 | 2.4 |
| \(= \arctan(k - 2) - \frac{\pi}{4}\) or \(\frac{\pi}{2} - \frac{\pi}{4}\) | B1 | 1.1 |
| \(\Rightarrow\) integral \(= \dfrac{\pi}{4}\) | B1 | 2.2a |
| [5] |
Notes
M1: completing the square, (ignore limits)
A1: \([\arctan(x - 2)]\) or \([\arctan u]\) if \(u = x - 2\) (ignore limits)
E1: Clear limit argument