Edexcel A-Level Mathematics AS P1.5 Integration Questions

$$\frac{(3x+2)^2}{4\sqrt{x}}=\frac{9}{4}x^{3/2}+3x^{1/2}+x^{-1/2}.$$
Therefore
$$\int \frac{(3x+2)^2}{4\sqrt{x}}\,dx =\frac{9}{10}x^{5/2}+2x^{3/2}+2x^{1/2}+c.$$

M1 Attempts to use $\mathrm{f}^{\prime}(3)=2 \Rightarrow 2=9+3 a+b$ oe. The expression does not need to be simplified so embedded values scores this mark. Condone slips when squaring but do not allow $\mathrm{f}^{\prime}(3)=-2$

M1 Attempts to integrate and achieves $(\mathrm{f}(x)=) \ldots x^{3}+\ldots a x^{2}+b x+(c)$ with or without the $+c. a, b \neq 0$
dM1 Uses y intercept =-8 and (3,-2) in their $(\mathrm{f}(x)=) \ldots x^{3}+\ldots a x^{2}+b x+c$. The values embedded in the expression are sufficient. It is dependent on the previous method mark only.
Alternatively uses (3,-2) in $(\mathrm{f}(x)=) \ldots x^{3}+\ldots a x^{2}+b x-8$
Beware: they may also substitute in b=-7-3 a at some point to achieve an equation in a only.
A1 Correct unsimplified equation in a and $b \quad-2=9+\frac{9}{2} a+3 b-8$.
Note: simplified becomes 9 a+6 b=-6
Alternatively they may have a correct unsimplified equation in a only: $-2=9+\frac{9}{2} a+3(-7-3 a)-8$ oe eg 4.5 a=-18
ddM1 Dependent upon all previous M's.
It is for solving their two equations with c=-8 to find values for a and b or solving their equation in a and then substituting in to b=-7-3 a to find a value for b

Don't be too concerned by the process. A calculator method is acceptable.
A1 $\frac{1}{3} x^{3}-2 x^{2}+5 x-8$ as their final answer. Do not isw eg $\frac{1}{3} x^{3}-2 x^{2}+5 x-8 \rightarrow x^{3}-6 x^{2}+15 x-24 y=\ldots$ or $\mathrm{f}(x)=\ldots$ are not required. Do not allow $5 x^{1}$.