C4 June 2005 Q3
3.
(a) Express \(\dfrac{5x + 3}{(2x - 3)(x + 2)}\) in partial fractions. (3)
(b) Hence find the exact value of \(\displaystyle\int_2^6 \frac{5x + 3}{(2x - 3)(x + 2)}\,\mathrm{d}x\), giving your answer as a single logarithm. (5)
| Scheme | Marks |
|---|---|
| \(\dfrac{5x + 3}{(2x - 3)(x + 2)} = \dfrac{A}{2x - 3} + \dfrac{B}{x + 2}\) | |
| \(5x + 3 = A(x + 2) + B(2x - 3)\) | |
| Substituting \(x = -2\) or \(x = \frac{3}{2}\) and obtaining \(A\) or \(B\); or equating coefficients and solving a pair of simultaneous equations to obtain \(A\) or \(B\). | M1 |
| \(A = 3, B = 1\) | A1, A1 |
| (3) |
Notes
If the cover-up rule is used, give M1 A1 for the first of \(A\) or \(B\) found, A1 for the second.
| Scheme | Marks |
|---|---|
| \(\displaystyle\int \frac{5x + 3}{(2x - 3)(x + 2)}\,\mathrm{d}x = \frac{3}{2}\ln(2x - 3) + \ln(x + 2)\) | M1 A1ft |
| \(\Big[\ \ldots\ \Big]_2^6 = \dfrac{3}{2}\ln 9 + \ln 2\) | M1 A1 |
| \(= \ln 54\) cao | A1 |
| (5) | |
| (8 marks) |