A2 June 2022 Paper 1 Q7
7
Hence determine the exact value of \(\displaystyle\int_3^{\infty} \frac{x^2 + 18}{x^2\left(x^2 + 9\right)}\,\mathrm{d}x\). [6]
| Scheme | Marks | AO |
|---|---|---|
| \(x^2 + 18 \equiv Ax\left(x^2 + 9\right) + B\left(x^2 + 9\right) + (Cx + D)x^2\) | B1 | 1.1 |
| e.g. \(x = 0 \Rightarrow 9B = 18 \Rightarrow B = 2\) \(x = 1 \Rightarrow 10A + 10B + C + D = 19\) \(x = -1 \Rightarrow -10A + 10B - C + D = 19\) \(\Rightarrow 10B + D = 19 \Rightarrow D = -1\) \(x = 3\mathrm{i} \Rightarrow -9(D + 3C\mathrm{i}) = 9 \Rightarrow C = 0\) \(\Rightarrow 10A + 20 - 1 = 19 \Rightarrow A = 0\) | M1 | 1.1 |
| A1 | 1.1 | |
| i.e. \(A = 0\), \(B = 2\), \(C = 0\), \(D = -1\) | A1 | 1.1 |
| [4] |
Notes
B1: Correct multiplying out of fractions
M1: Any substitutions to get a set of (at least) four simultaneous equations solvable for \(A, B, C\) and \(D\).
Or equating coefficients which gives \(A + C = 0, B + D = 1, 9A = 0, 9B = 18\).
A1: Any two coefficients correct.
A1: All four coefficients correct.
SC B1 after M0 if one or more coefficients are correct.
| Scheme | Marks | AO |
|---|---|---|
| DR \(\displaystyle\int\left(\frac{2}{x^2} - \frac{1}{x^2 + 9}\right)\mathrm{d}x = -\frac{2}{x} - \frac{1}{3}\tan^{-1}\frac{x}{3}\ (+c)\) | M1 A1 | 1.1 1.1 |
| \(\displaystyle\Rightarrow \int_3^{\infty}\left(\frac{2}{x^2} - \frac{1}{x^2 + 9}\right)\mathrm{d}x\) \(\displaystyle = \lim_{k \to \infty}\left(-\left(\frac{2}{k} - \frac{2}{3}\right) - \frac{1}{3}\left(\tan^{-1}\frac{k}{3} - \tan^{-1}1\right)\right)\) | M1 | 1.1 |
| \(\displaystyle = \frac{2}{3} - \lim_{k \to \infty}\frac{2}{k} + \frac{\pi}{12} - \frac{1}{3}\lim_{k \to \infty}\left(\tan^{-1}\frac{k}{3}\right)\) | A1 | 2.1 |
| \(= \dfrac{2}{3} - 0 + \dfrac{\pi}{12} - \dfrac{1}{3} \times \dfrac{\pi}{2}\) | A1 | 2.1 |
| \(= \dfrac{2}{3} - \dfrac{\pi}{12}\) | A1 | 1.1 |
| [6] |
Notes
M1: integration including a \(\tan^{-1}\) term
A1: ft their part (a)
M1: Use of limiting process on their integrated function. Ignore notation for limits
A1: for \(\displaystyle\lim_{k \to \infty}\left(\frac{1}{k}\right) = 0\), or as \(k \to \infty\), \(\frac{1}{k} \to 0\), A0 for eg. \(\frac{1}{\infty} = 0\).
A1: \(\displaystyle\lim_{k \to \infty}\left(\tan^{-1}\frac{k}{c}\right) = \frac{1}{2}\pi\), or as \(k \to \infty\), \(\tan^{-1}\frac{k}{c} \to \frac{1}{2}\pi\), A0 for eg. \(\tan^{-1}\infty = \frac{1}{2}\pi\). In both cases must see some evidence of the limiting process.