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Question
The value of \[\int\frac{1}{x + x \log x} dx\] is
Options
1 + log x
x + log x
x log (1 + log x)
log (1 + log x)
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Solution
log (1 + log x)
\[\text{Let }I = \int\frac{dx}{x + x \log x}\]
\[ \Rightarrow \int\frac{dx}{x \left( 1 + \log x \right)}\]
\[\text{Putting }1 + \log x = t\]
\[\Rightarrow \frac{1}{x} dx = dt\]
\[ \therefore I = \int\frac{dt}{t}\]
\[ = \ln \left| t \right| + C\]
\[ = \ln \left| 1 + \log x \right| + C\]
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