Moderate -0.3 This is a straightforward application of integration by parts with a standard integrand x·ln(x). The technique is routine (u=ln x, dv=x dx), limits are simple integers, and it's a single-step problem worth modest marks. Slightly easier than average due to its directness, though integration by parts itself is a core A-level skill.
| Content | Mark | Guidance |
|---------|------|----------|
| **EITHER:** State first step of the form $kx^2 \ln x \pm \int kx^2 \cdot \frac{1}{x} dx$ | M1 | |
| Obtain correct first step i.e. $\frac{1}{2}x^2 \ln x - \int \frac{1}{2}x dx$ | A1 | |
| Complete a second integration and substitute both limits correctly | M1 | |
| Obtain correct answer $2 \ln 2 - \frac{3}{4}$, or exact two-term equivalent | A1 | |
| **OR:** State first step of the form $I = x(x\ln x \pm x) \pm \int (x\ln x \pm x)dx$ | M1 | |
| Obtain correct first step i.e. $I = x(x\ln x - x) - I + \int x dx$ | A1 | |
| Complete a second integration and substitute both limits correctly | M1 | |
| Obtain correct answer $2 \ln 2 - \frac{3}{4}$, or exact two-term equivalent | A1 | Max 4 marks |
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