C4 June 2009 Q3
3. \[\mathrm{f}(x) = \frac{4 - 2x}{(2x + 1)(x + 1)(x + 3)} = \frac{A}{2x + 1} + \frac{B}{x + 1} + \frac{C}{x + 3}\]
(a) Find the values of the constants \(A\), \(B\) and \(C\). (4)
(b)
(i) Hence find \(\displaystyle\int \mathrm{f}(x)\,\mathrm{d}x\). (3)
(ii) Find \(\displaystyle\int_0^2 \mathrm{f}(x)\,\mathrm{d}x\) in the form \(\ln k\), where \(k\) is a constant. (3)
| Scheme | Marks |
|---|---|
| \(\mathrm{f}(x) = \dfrac{4 - 2x}{(2x + 1)(x + 1)(x + 3)} = \dfrac{A}{2x + 1} + \dfrac{B}{x + 1} + \dfrac{C}{x + 3}\) | |
| \(4 - 2x = A(x + 1)(x + 3) + B(2x + 1)(x + 3) + C(2x + 1)(x + 1)\) | M1 |
| A method for evaluating one constant | M1 |
| \(x \to -\tfrac{1}{2},\quad 5 = A\left(\tfrac{1}{2}\right)\left(\tfrac{5}{2}\right) \Rightarrow A = 4\) any one correct constant | A1 |
| \(x \to -1,\quad 6 = B(-1)(2) \Rightarrow B = -3\) \(x \to -3,\quad 10 = C(-5)(-2) \Rightarrow C = 1\) all three constants correct | A1 |
| (4) |
| Scheme | Marks |
|---|---|
| (i) \(\displaystyle\int\left(\frac{4}{2x + 1} - \frac{3}{x + 1} + \frac{1}{x + 3}\right)\mathrm{d}x\) | |
| \(= \dfrac{4}{2}\ln(2x + 1) - 3\ln(x + 1) + \ln(x + 3) + C\) A1 two ln terms correct | M1 A1ft |
| All three ln terms correct and “\(+C\)”; ft constants | A1ft |
| (3) | |
| (ii) \(\Big[2\ln(2x + 1) - 3\ln(x + 1) + \ln(x + 3)\Big]_0^2\) | |
| \(= (2\ln 5 - 3\ln 3 + \ln 5) - (2\ln 1 - 3\ln 1 + \ln 3)\) | M1 |
| \(= 3\ln 5 - 4\ln 3\) | |
| \(= \ln\left(\dfrac{5^3}{3^4}\right)\) | M1 |
| \(= \ln\left(\dfrac{125}{81}\right)\) | A1 |
| (3) | |
| (10 marks) |