A2 June 2019 Paper 1 Q15
15 In this question you must show detailed reasoning.
Show that \(\displaystyle\int_{\frac{3}{4}}^{\frac{3}{2}} \frac{1}{\sqrt{4x^2 - 4x + 2}}\,\mathrm{d}x = \frac{1}{2}\ln\left(\frac{3 + \sqrt{5}}{2}\right)\). [8]
| Scheme | Marks | AO |
|---|---|---|
| DR \(\displaystyle\int_{\frac{3}{4}}^{\frac{3}{2}} \frac{1}{\sqrt{4x^2 - 4x + 2}}\,\mathrm{d}x = \int_{\frac{3}{4}}^{\frac{3}{2}} \frac{1}{\sqrt{(2x - 1)^2 + 1}}\,\mathrm{d}x\) | M1 | 3.1a |
| \(= \left[\dfrac{1}{2}\operatorname{arsinh}(2x - 1)\right]_{\frac{3}{4}}^{\frac{3}{2}}\) | M1 A1 | 1.1b 1.1b |
| (3) | ||
| \(= \dfrac{1}{2}\left[\operatorname{arsinh}(2) - \operatorname{arsinh}\left(\dfrac{1}{2}\right)\right]\) \(= \dfrac{1}{2}\left[\ln(2 + \sqrt{5}) - \ln\left(\dfrac{1}{2} + \dfrac{\sqrt{5}}{2}\right)\right]\) | M1 A1 | 1.1b |
| \(= \dfrac{1}{2}\ln\dfrac{2(\sqrt{5} + 2)}{\sqrt{5} + 1}\) | M1 | 2.1 |
| \(= \dfrac{1}{2}\ln\dfrac{2(\sqrt{5} + 2)(\sqrt{5} - 1)}{(\sqrt{5} + 1)(\sqrt{5} - 1)}\) | M1 | 2.1 |
| \(= \dfrac{1}{2}\ln\dfrac{(\sqrt{5} + 3)}{2}\,*\) | A1cao | 2.1 |
| (5) | ||
| [8] |
Notes
M1: attempt to complete the square; or \(\displaystyle\frac{1}{2}\int_{\frac{3}{4}}^{\frac{3}{2}} \frac{1}{\sqrt{(x - 1/2)^2 + 1/4}}\,\mathrm{d}x\)
M1: \(\operatorname{arsinh}(2x - 1)\) (oe); \(= \left[\frac{1}{2}\operatorname{arsinh} u\right]_{\frac{1}{2}}^{2}\) if \(u = 2x - 1\)
A1: \(\times \frac{1}{2}\) oe e.g. ln form
M1: \(\operatorname{arsinh} x = \ln\left(x + \sqrt{x^2 + 1}\right)\) (used)
A1: correct expression
M1: combining lns
M1: rationalizing denominator (must be seen)
A1cao: NB AG