FP3 June 2010 Q2
2. Use calculus to find the exact value of \(\displaystyle\int_{-2}^{1} \frac{1}{x^2 + 4x + 13}\,\mathrm{d}x\). (5)
| Scheme | Marks |
|---|---|
| \(x^2 + 4x + 13 = (x + 2)^2 + 9\) | B1 |
| \(\displaystyle\int \dfrac{1}{(x + 2)^2 + 9}\,\mathrm{d}x = \dfrac{1}{3}\arctan\left(\dfrac{x + 2}{3}\right)\) | M1 A1 |
| \(\left[\dfrac{1}{3}\arctan\left(\dfrac{x + 2}{3}\right)\right]_{-2}^{1} = \dfrac{1}{3}\left(\arctan 1 - \arctan 0\right)\) | M1 |
| \(= \dfrac{\pi}{12}\) | A1 |
| (5) | |
| (5 marks) |