FP3 June 2014 (R) Q2
2. \[9x^2 + 6x + 5 \equiv a(x + b)^2 + c\]
(a) Find the values of the constants \(a\), \(b\) and \(c\). (3)
Hence, or otherwise, find
(b) \(\displaystyle\int \frac{1}{9x^2 + 6x + 5}\,\mathrm{d}x\) (2)
(c) \(\displaystyle\int \frac{1}{\sqrt{9x^2 + 6x + 5}}\,\mathrm{d}x\) (2)
| Scheme | Marks |
|---|---|
| \(9x^2 + 6x + 5 \equiv a(x + b)^2 + c\) | |
| \(a = 9,\ b = \tfrac{1}{3},\ c = 4\) | B1, B1, B1 |
| (3) |
| Scheme | Marks |
|---|---|
| \(\displaystyle\int \frac{1}{9(x + \frac{1}{3})^2 + 4}\,\mathrm{d}x = \frac{1}{6}\arctan\left(\frac{3x + 1}{2}\right)(+c)\) M1: \(k\arctan\left(\dfrac{x + \text{"}\frac{1}{3}\text{"}}{\sqrt{\frac{\text{"}4\text{"}}{\text{"}9\text{"}}}}\right)\) A1: \(\dfrac{1}{6}\arctan\left(\dfrac{3x + 1}{2}\right)\) oe | M1A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(\displaystyle\int \frac{1}{\sqrt{9(x + \frac{1}{3})^2 + 4}}\,\mathrm{d}x = \frac{1}{3}\mathrm{arsinh}\left(\frac{3x + 1}{2}\right)(+c)\) M1: \(k\,\mathrm{arsinh}\left(\dfrac{x + \text{"}\frac{1}{3}\text{"}}{\sqrt{\frac{\text{"}4\text{"}}{\text{"}9\text{"}}}}\right)\) A1: \(\dfrac{1}{3}\mathrm{arsinh}\left(\dfrac{3x + 1}{2}\right)\) oe Allow \(\dfrac{1}{\sqrt{9}}\) | M1A1 |
| (2) | |
| (7 marks) |