## Part (a)
$\frac{3 + 13x - 6x^2}{2x - 3} = Ax + B + \frac{C}{2x - 3}$ | M1 | e.g. using $x = \frac{3}{2}$ in an attempt to find $C$ or forming simultaneous equations and attempt to solve.

$3 + 13x - 6x^2 = Ax(2x - 3) + B(2x - 3) + C$

Correct above equation and attempt to find one of $A$, $B$ or $C$ or an attempt at long division | |

$A = -3$ | A1 |

$B = 2$ | A1 |

$C = 9$ | A1 | Total: 4

If long division is used award M1 once $-3x + \cdots$ has been obtained but only award the A marks once the values are clearly identified or it is written in the required form of $Ax + B + \frac{C}{2x - 3}$

Alternative method of division $\frac{3+13x-6x^2}{2x-3} = \frac{-3x(2x-3)-9x+13x+3}{2x-3} = -3x + \frac{4x+3}{2x-3}$ | M1 for $-3x + \cdots$,

For the A marks, $A$, $B$ and $C$ must be clearly identified or seen in the required form of $Ax + B + \frac{C}{2x-3}$

NMS scores B2 for one correct value, B3 for 2 correct values and B4 for all three correct values

## Part (b)
$\int\frac{3+13x-6x^2}{2x-3}dx = \int(-3x + 2 + \frac{9}{2x-3})dx$ | M1 |

$= px^2 + qx + r\ln(2x - 3)$ | |

$= -\frac{3}{2}x^2 + 2x + \frac{9}{2}\ln(2x - 3)$ | A1ft | $\frac{4}{3}x^2 + Bx + \frac{5}{2}\ln(2x - 3)$

Correct use of $F(6) - F(3)$

$= \left[-\frac{3}{2}\cdot 6^2 + 2.6 + \frac{9}{2}\ln(12 - 3)\right]$ | m1 | Correct substitution of limits for their $p, q$ and $r$.

$- \left[-\frac{3}{2}\cdot 3^2 + 2.3 + \frac{9}{2}\ln(6 - 3)\right]$

$= -\frac{69}{2} + \frac{9}{2}\ln3$ | A1 | Total: 4 | OE

The M1 A1ft and m1 can be earned even if left in terms of $A, B$ and $C$ or if 'invented' value(s) for $A, B$ and $C$ are used.

Condone missing brackets from the $\ln(2x - 3)$ term for the M1 mark but only award the A1ft mark if they have clearly recovered; PI by sight of $\ln9$ or $\ln3$ after using the limits or a correct final answer.

Treat a decimal answer (should be–29.55 ...) after a correct exact form as ISW but award A0 if an exact answer is not seen.

Total | | 8

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