Moderate -0.3 This is a straightforward integration by parts question with a standard form (x^n·ln(x)). Part (a) requires one application of integration by parts with clear choices for u and dv, followed by routine integration. Part (b) is direct substitution of limits. While it requires careful algebraic manipulation, it follows a well-practiced technique with no conceptual surprises, making it slightly easier than average for C4.
\(-\frac{1}{2x^2}\ln x + \frac{1}{2}\left(-\frac{1}{2x^2}\right)\{+c\}\)
A1
Correct answer with/without \(+c\)
Question 2(b):
Answer
Marks
Guidance
Answer/Working
Marks
Guidance
\(\left[-\frac{1}{2x^2}\ln x - \frac{1}{4x^2}\right]_1^2\) applied with limits 2 and 1
M1
Applies limits of 2 and 1 to part (a) answer, subtracts correct way round
\(\frac{3}{16} - \frac{1}{8}\ln 2\) or \(\frac{3}{16} - \ln 2^{\frac{1}{8}}\) or \(\frac{1}{16}(3-2\ln 2)\) or equivalent, awrt 0.1
A1
Two term exact answer required; fraction terms must be combined
Note: Decimal answer is 0.100856... Award final A0 in (b) for candidate who achieves awrt 0.1 when part (a) answer is incorrect.
## Question 2(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| In the form $\frac{\pm\lambda}{x^2}\ln x \pm \int \mu\frac{1}{x^2}\cdot\frac{1}{x}$ | M1 | Integration by parts applied in correct form |
| $\frac{-1}{2x^2}\ln x$ simplified or un-simplified | A1 | |
| $-\int\frac{-1}{2x^2}\cdot\frac{1}{x}$ simplified or un-simplified | A1 | |
| $\pm\int\mu\frac{1}{x^2}\cdot\frac{1}{x} \rightarrow \pm\beta x^{-2}$ | dM1 | Depends on previous M1 |
| $-\frac{1}{2x^2}\ln x + \frac{1}{2}\left(-\frac{1}{2x^2}\right)\{+c\}$ | A1 | Correct answer with/without $+c$ |
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## Question 2(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\left[-\frac{1}{2x^2}\ln x - \frac{1}{4x^2}\right]_1^2$ applied with limits 2 and 1 | M1 | Applies limits of 2 and 1 to part (a) answer, subtracts correct way round |
| $\frac{3}{16} - \frac{1}{8}\ln 2$ or $\frac{3}{16} - \ln 2^{\frac{1}{8}}$ or $\frac{1}{16}(3-2\ln 2)$ or equivalent, awrt 0.1 | A1 | Two term exact answer required; fraction terms must be combined |
**Note:** Decimal answer is 0.100856... Award final A0 in (b) for candidate who achieves awrt 0.1 when part (a) answer is incorrect.
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