Question

Evaluate each of the following integral:

\[\int_e^{e^2} \frac{1}{x\log x}dx\]

Answer 1

\[\int_e^{e^2} \frac{1}{x\log x}dx\]
\[ = \int_e^{e^2} \frac{\frac{1}{x}}{\log x}dx\]
\[ = \left.\log\left( \log x \right)\right|_e^{e^2} ...............\left[ \int\frac{f'\left( x \right)}{f\left( x \right)}dx = \log f\left( x \right) + C \right]\]
\[ = \log\left( \log e^2 \right) - \log\left( \log e \right)\]
\[ = \log\left( 2\log e \right) - \log\left( \log e \right) \]
\[ = \log2 - \log1 ................\left( \log e  = 1 \right)\]
\[ = \log2 - 0\]
\[ = \log2\]