Moderate -0.3 This is a straightforward substitution question with clear guidance (substitution given explicitly). Students must express 4x-1 in terms of u, adjust limits, and integrate a simple power. It's slightly easier than average because the substitution is provided and the algebra is routine, though it requires careful handling of the numerator transformation.
Correct result e.g. \(\frac{du}{dx} = 2\) or \(du = 2\,dx\)
A1
Indef. integral in terms of \(u = \frac{1}{2}\int\frac{2u-3}{u^5}\,du\)
A1
Must be completely in terms of \(u\)
Integrate to \(\frac{u^{-3}}{-3} - \frac{3u^{-4}}{-8}\) oe
A1A1
Or (using 'by parts') \(\frac{(2u-3)u^{-4}}{-8} - \frac{u^{-3}}{12}\); award B1,B1 for \(\frac{4u^{-3}}{-3} - \frac{3u^{-4}}{-2}\) or other variants listed
Use correct variable and correct values for limits
M1
Provided minimal attempt at \(\int f(u)\,du\) made
\(= \frac{-23}{384}\) oe \((-0.059895\ldots)\)
A1
Accept decimal only if minimum of first 3 marks scored
[7]
# Question 6:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt to connect $du$ and $dx$ | M1 | or find $\frac{du}{dx}$ or $\frac{dx}{du}$ |
| Correct result e.g. $\frac{du}{dx} = 2$ or $du = 2\,dx$ | A1 | |
| Indef. integral in terms of $u = \frac{1}{2}\int\frac{2u-3}{u^5}\,du$ | A1 | Must be completely in terms of $u$ |
| Integrate to $\frac{u^{-3}}{-3} - \frac{3u^{-4}}{-8}$ oe | A1A1 | Or (using 'by parts') $\frac{(2u-3)u^{-4}}{-8} - \frac{u^{-3}}{12}$; award B1,B1 for $\frac{4u^{-3}}{-3} - \frac{3u^{-4}}{-2}$ or other variants listed |
| Use correct variable and correct values for limits | M1 | Provided minimal attempt at $\int f(u)\,du$ made |
| $= \frac{-23}{384}$ oe $(-0.059895\ldots)$ | A1 | Accept decimal only if minimum of first 3 marks scored |
| **[7]** | | |
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