Question

Evaluate : \[\int\frac{1}{x(1 + \log x)} \text{ dx}\]

Answer 1

I = \[\int\frac{1}{x\left( 1 + \log x \right)} \text{ dx }\]

Let (1 + log x) = t

or,\[ \frac{1}{x}dx = dt\]
\[ \Rightarrow I = \int\frac{1}{t}dt\]

\[\Rightarrow I = \text{ log }\left| t \right| + C\]
\[ \therefore I = \text{ log}\left| 1 + \log x \right| + C\]