A2 June 2021 Paper 1 Q10
10 Evaluate the improper integral
\[\int_0^8 \ln x\,\mathrm{d}x\]showing the limiting process. [6 marks]
| Scheme | Marks | AO |
|---|---|---|
| Defines the improper integral as a limit | E1 | 2.4 |
| Selects and uses the method of integration by parts. Implied by stating and using the formula for the integral of \(\ln x\) | M1 | 3.1a |
| Obtains the correct integral with or without \(c\). Condone no limits | A1 | 1.1b |
| Substitutes 8 correctly into their two-term expression for the integral | M1 | 1.1a |
| Applies the limiting process correctly, using \(\lim_{h \to 0}\{h\ln(h)\} = 0\) This does not have to be stated explicitly | M1 | 2.2a |
| Obtains correct value OE, explicitly stating \(\lim_{h \to 0}\{h\ln(h)\} = 0\) NMS = 0 | A1 | 1.1b |
| (6 marks) |
Typical solution
\[\int_0^8 \ln(x)\,\mathrm{d}x = \lim_{h \to 0}\int_h^8 \ln(x)\,\mathrm{d}x\]\[u = \ln x \qquad v^{\prime} = 1\]\[u^{\prime} = \frac{1}{x} \qquad v = x\]\[\int_0^8 \ln(x)\,\mathrm{d}x = \lim_{h \to 0}\left(\left[x\ln(x)\right]_h^8 - \int_h^8 1\,\mathrm{d}x\right)\]\[= \lim_{h \to 0}\left(\left[x\ln(x) - x\right]_h^8\right)\]\[= \{8\ln(8) - 8\} - \lim_{h \to 0}\{h\ln(h) - h\}\]As \(\lim_{h \to 0}\{h\ln(h)\} = 0\)
Then \(\displaystyle\int_0^8 \ln(x)\,\mathrm{d}x = 8\ln(8) - 8\)