A2 June 2025 Paper 1 Q10
10 In this question you must show detailed reasoning.
Evaluate \(\displaystyle\int_0^{\frac{1}{2}} \frac{2}{x^2 - x + 1}\,\mathrm{d}x\). Give your answer in exact form. [4]
| Scheme | Marks | AO |
|---|---|---|
| DR \(= 2\displaystyle\int_0^{\frac{1}{2}} \frac{1}{\left(x - \frac{1}{2}\right)^2 + \frac{3}{4}}\,\mathrm{d}x\) | M1* | 3.1a |
| \(= \left[\frac{4}{\sqrt{3}}\arctan\left(\frac{x - \frac{1}{2}}{\frac{\sqrt{3}}{2}}\right)\right]_0^{\frac{1}{2}}\) or \(\left[\frac{4}{\sqrt{3}}\arctan\left(\frac{2x - 1}{\sqrt{3}}\right)\right]_0^{\frac{1}{2}}\) | A1 | 2.1 |
| \(= \frac{4}{\sqrt{3}}\left(\arctan(0) - \arctan\left(\frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}}\right)\right)\) | M1dep | 2.1 |
| \(= \dfrac{2\pi}{3\sqrt{3}}\) | A1cao | 1.1 |
| [4] |
Notes
M1*: completing the square correctly on \(x^2 - x + 1\); need not be in integral but must be seen
A1: oe; condone missing or incorrect limits. Could be in terms of another variable if substitution used for integration, e.g. if \(u = x - \frac{1}{2}\) then \(\left[\frac{4}{\sqrt{3}}\arctan\left(\frac{u}{\frac{\sqrt{3}}{2}}\right)\right]_{-\frac{1}{2}}^{0}\) or if \(\frac{\sqrt{3}}{2}\tan u = x - \frac{1}{2}\) then \(\left[\frac{4}{\sqrt{3}}u\right]_{-\frac{\pi}{6}}^{0}\). Condone \(\left[\frac{4}{\sqrt{3}}\arctan\left(\frac{x}{\frac{\sqrt{3}}{2}}\right)\right]_{-\frac{1}{2}}^{0}\) if the correct substitution has been clearly made. Must be seen.
M1dep: using correct limits correctly, including following any substitution. Substitution into (or evaluation of) any non-zero terms must be seen
A1cao: or simplified, exact equivalent. www.