| Date | November 2015 | Marks available | 4 | Reference code | 15N.1.hl.TZ0.5 |
| Level | HL only | Paper | 1 | Time zone | TZ0 |
| Command term | Find and Use | Question number | 5 | Adapted from | N/A |
Question
Use the substitution \(u = \ln x\) to find the value of \(\int_{\text{e}}^{{{\text{e}}^2}} {\frac{{{\text{d}}x}}{{x\ln x}}} \).
Markscheme
METHOD 1
\(\int_{\text{e}}^{{{\text{e}}^2}} {\frac{{{\text{d}}x}}{{x\ln x}}} = \left[ {\ln (\ln x)} \right]_{\text{e}}^{{{\text{e}}^2}}\) (M1)A1
\( = \ln (\ln {{\text{e}}^2}) - \ln (\ln {\text{e}})\;\;\;( = \ln 2 - \ln 1)\) (A1)
\( = \ln 2\) A1
[4 marks]
METHOD 2
\(u = \ln x,{\text{ }}\frac{{{\text{d}}u}}{{{\text{d}}x}} = \frac{1}{x}\) M1
\( = \int_1^2 {\frac{{{\text{d}}u}}{u}} \) A1
Note: Condone absent or incorrect limits here.
\( = [\ln u]_1^2\) or equivalent in \(x( = \ln 2 - \ln 1)\) (A1)
\( = \ln 2\) A1
[4 marks]
Examiners report
[N/A]
