June 2025 Paper 2 Q8
8
In this question you must show detailed reasoning.
Use integration by parts to find the exact value of \(\displaystyle\int_0^{\frac{1}{2}} \ln(3 - 2x)\,\mathrm{d}x\). [5]
| Scheme | Marks | AO |
|---|---|---|
| DR | ||
| \(x\ln(3 - 2x) + \displaystyle\int \frac{2x}{3 - 2x}\,\mathrm{d}x\) | M1* | 3.1a |
| \(= x\ln(3 - 2x) + \displaystyle\int \left(-1 + \frac{3}{3 - 2x}\right)\mathrm{d}x\) | M1** dep* | 3.1a |
| \(= x\ln(3 - 2x) - x - \frac{3}{2}\ln(3 - 2x)\,[+c]\) | A1 | 1.1 |
| \(\left(\frac{1}{2}\ln 2 - \frac{1}{2} - \frac{3}{2}\ln 2\right) - \left(-\frac{3}{2}\ln 3\right)\) oe | M1 dep** | 2.1 |
| \(= \frac{3}{2}\ln 3 - \ln 2 - \frac{1}{2}\) or \(\ln\left(\frac{3\sqrt{3}}{2}\right) - \frac{1}{2}\) oe | A1 | 1.1 |
| [5] |
Notes
Condone missing ‘\(\mathrm{d}x\)’ or ‘\(\mathrm{d}u\)’ throughout all methods.
M1*: Attempt integration by parts – must have correct parts, possibly with \(\int x \times (-2) \times \frac{1}{3 - 2x}\,\mathrm{d}x\) not yet simplified. Allow a sign error or missing (−)2. No limits needed for M1M1A1 (first 3 marks)
M1** dep*: Attempt partial fractions; allow one sign error
M1 dep**: Attempt substitute correct limits \(0, \frac{1}{2}\) and subtract (allow one slip). FT their integral. Dependent on both previous M marks.
A1: For any correct exact form, including \(\left(\frac{1}{2}\ln 2 - \frac{1}{2} - \frac{3}{2}\ln 2\right) - \left(-\frac{3}{2}\ln 3\right)\), need not be simplified. ISW.
Answer \(0.45477\ldots\) scores A0 unless exact answer also seen
Alternative method using substitution following correct setup of integration by parts
| Scheme | Marks | AO |
|---|---|---|
| \(x\ln(3 - 2x) + \displaystyle\int \frac{2x}{3 - 2x}\,\mathrm{d}x\) | M1* | |
| \(= x\ln(3 - 2x) + \displaystyle\int \frac{3 - u}{u}\left(-\frac{1}{2}\right)\mathrm{d}u\) | M1** dep* | |
| \(= \dfrac{3 - u}{2}\ln(u) - \dfrac{1}{2}\left[3\ln u - u\right]\) | A1 | |
| \(= \frac{1}{2}\ln 2 - \frac{1}{2}(3\ln 2 - 2) + \frac{1}{2}(3\ln 3 - 3)\) | M1 dep** | |
| \(= \frac{3}{2}\ln 3 - \ln 2 - \frac{1}{2}\) or \(\ln\left(\frac{3\sqrt{3}}{2}\right) - \frac{1}{2}\) oe | A1 |
If candidates apply integration by parts and then use substitution, apply this method. (NB candidates can switch from this back to the main method if they revert correctly from \(u\) to \(x\) so look for limits used correctly in either \(x\) or \(u\) as appropriate).
M1*: Attempt integration by parts – must have correct parts, possibly with \(\int x \times (-2) \times \frac{1}{3 - 2x}\,\mathrm{d}x\) not yet simplified. Allow a sign error or missing ‘2’. No limits needed for M1M1A1 (first 3 marks)
M1** dep*: Using e.g. \(u = 3 - 2x\), allow one sign error.
A1: Ignore any additional constant term e.g. \(+c\) or \(+k\) for any \(k\).
M1 dep**: Attempt to substitute transformed limits 3, 2 and subtract (condone swapped limits). FT their integral. Dep on both previous M marks.
A1: For any correct exact form, including \(\left(\frac{1}{2}\ln 2 - \frac{1}{2} - \frac{3}{2}\ln 2\right) - \left(-\frac{3}{2}\ln 3\right)\), need not be simplified. ISW.
Answer \(0.45477\ldots\) scores A0 unless exact answer also seen
Alternative method for Q8 applying substitution first
| Scheme | Marks | AO |
|---|---|---|
| M1* | ||
| \(-\dfrac{1}{2}\displaystyle\int_3^2 \ln u\,\mathrm{d}u\) | A1 | |
| \(= -\dfrac{1}{2}\left[u\ln u - \displaystyle\int u\frac{1}{u}\,\mathrm{d}u\right]\) | M1 dep* | |
| \(= -\dfrac{1}{2}\left[u\ln u - u\right]_3^2\) | A1 | |
| \(= -\ln 2 + 1 + \frac{3}{2}\ln 3 - \frac{3}{2}\) oe | A1 |
If candidates start by applying a substitution, then apply this method.
M1*: Attempting to integrate by substitution using e.g. \(u = 3 - 2x\) (no limits required for this mark). Allow a sign error or missing \(-\frac{1}{2}\).
A1: Fully correct integral with transformed limits (may be seen later, condone swapped limits).
M1 dep*: Use of integration by parts. This mark is not implied by later correct working – must see this form or e.g. \(\ln u\), \(\frac{1}{u}\), \(u\) and 1 as setup for integration by parts. Ignore limits for this mark. Condone a sign error or missing \(-\frac{1}{2}\).
A1: oe but must be correct
A1: For any correct form (as above). ISW
NB candidates who do not show evidence of integration by parts used anywhere can score max. 2/5 under this method.
Answer \(0.45477\ldots\) scores A0 unless exact answer also seen
Alternative method for Q8 assuming formula for \(\int \ln x\,\mathrm{d}x\) (without substitution) (Only 2 marks, since this Q is DR and required integration by parts)
| Scheme | Marks | AO |
|---|---|---|
| \(-\dfrac{1}{2}\left[(3 - 2x)\ln(3 - 2x) - (3 - 2x)\right]_0^{\frac{1}{2}}\) | M1 | |
| \(= \frac{1}{2}\ln 2 - \frac{1}{2} - \frac{3}{2}\ln 2 + \frac{3}{2}\ln 3\) oe | A1 |
If no evidence of integration by parts, or of substitution, then apply this method.
M1: Allow \(k\left[(3 - 2x)\ln(3 - 2x) - (3 - 2x)\right]_0^{\frac{1}{2}}\)
A1: ISW
Answer \(0.45477\ldots\) scores A0 unless exact answer also seen