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2 votes
2 votes
Which of the following expression evaluates to given integral
$$
\int \frac{\ln (\ln x)}{x \ln x} d x
$$
  1. $\dfrac{\ln x}{x}+C$
  2. $\frac{1}{2}(\ln \ln x)^{2}+C$
  3. $(\ln x)^{2}+C$
  4. $(\ln \ln x)+C$
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1 Answer

4 votes
4 votes
Let $u=\ln x$. Then $d u=\frac{1}{x} d x$. Thus we have
$$
\begin{aligned}
\int \frac{\ln \ln x}{x \ln x} d x &=\int \frac{\ln u}{u} d u \text { Let } v=\ln u, \text { then } d v=\frac{1}{u} d u \\
&=\int v d v \\
&=\frac{1}{2} v^{2}+C \\
&=\frac{1}{2}(\ln u)^{2}+C \\
&=\frac{1}{2}(\ln \ln x)^{2}+C
\end{aligned}
$$
Answer:

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