Standard +0.3 Part (i) is a standard integration by parts application with a straightforward substitution. Part (ii) involves routine partial fractions with a repeated linear factor and simple integration of the resulting terms. Both are textbook-standard techniques with no novel insight required, making this slightly easier than average for C4.
At least two terms correctly integrated; All three terms correctly integrated. Ignore absence of "\(+c\)"
[3]
[10]
Notes for 2.(i)
M1: Integration by parts is applied in the form \(\pm \lambda x\sin\left(\frac{x}{2}\right)-\int \mu\sin\left(\frac{x}{2}\right)\{dx\}\) (where \(\lambda \neq 0, \mu \neq 0\))
A1: \(2x\sin\left(\frac{x}{2}\right)-\int 2\sin\left(\frac{x}{2}\right)\{dx\}\) or equivalent. Can be un-simplified.
**(i)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int x\cos\left(\frac{x}{2}\right)dx$, with $u=x \Rightarrow \frac{du}{dx}=1$; $\frac{dv}{dx}=\cos\left(\frac{x}{2}\right) \Rightarrow v=2\sin\left(\frac{x}{2}\right)$ | M1 A1 | |
| $=2x\sin\left(\frac{x}{2}\right)-\int 2\sin\left(\frac{x}{2}\right)\{dx\}$ | A1 | |
| $=2x\sin\left(\frac{x}{2}\right)+4\cos\left(\frac{x}{2}\right)\{+c\}$ | A1 | |
| | | [3] |
**(ii)(a)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{1}{x^2(1-3x)}=\frac{A}{x}+\frac{B}{x^2}+\frac{C}{(1-3x)}$ | B1 | At least one of "$B$" or "$C$" correct. Breaks up their partial fraction correctly into **three terms** and both "$B$"$=1$ and "$C$"$=9$. |
| $B=1, C=9$ | B1 cso | See notes below. |
| $1 \equiv Ax(1-3x)+B(1-3x)+Cx^2$; $x=0, 1=B$; $x=\frac{1}{3}, 1=\frac{1}{9}C \Rightarrow C=9$; $x^2$ terms: $0=-3A+C \Rightarrow A=3$ | M1 | Writes down a **correct identity** and attempts to find the value of either one of "$A$", "$B$" or "$C$". |
| | A1 | Correct value for "$A$" which is found using a correct identity and follows from their partial fraction decomposition. |
| | | [4] |
**(ii)(b)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int \frac{1}{x^2(1-3x)}dx=\int\left[\frac{3}{x}+\frac{1}{x^2}+\frac{9}{(1-3x)}\right]dx$ | M1 | Either $\pm\frac{P}{x}\to\pm a\ln x$ or $\pm a\ln kx$; or $\pm\frac{Q}{x^2}\to\pm b x^{-1}$ or $\frac{R}{(1-3x)}\to\pm c\ln(1-3x)$ |
| $=3\ln x+\frac{x^{-1}}{(-1)}+\frac{9}{(-3)}\ln(1-3x)\{+c\}$ | A1ft | **At least two terms correctly integrated**; **All three terms correctly integrated**. Ignore absence of "$+c$" |
| | | [3] |
| | | [10] |
**Notes for 2.(i)**
M1: Integration by parts is applied in the form $\pm \lambda x\sin\left(\frac{x}{2}\right)-\int \mu\sin\left(\frac{x}{2}\right)\{dx\}$ (where $\lambda \neq 0, \mu \neq 0$)
A1: $2x\sin\left(\frac{x}{2}\right)-\int 2\sin\left(\frac{x}{2}\right)\{dx\}$ or equivalent. Can be un-simplified.
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