First slide

Methods of integration

Question

$\int [\log (\log x) + \frac{1}{{(\log x)}^{2}}] d x =$

Moderate

A

$\log (\log x) + \frac{x}{\log x} + c$
B

$x \log (\log x) + \frac{1}{\log x} + c$
C

$x \log (\log x) - \frac{x}{\log x} + c$
D

$- \log (\log x) - \frac{x}{\log x} + c$

Solution

$\int [\log (\log x) + \frac{1}{{(\log x)}^{2}}] d x = \int [\log (t) + \frac{1}{{(t)}^{2}}] e^{t} d t p u t \log x = t t h e n x = e^{t} \Rightarrow d x = e^{t} d t$
$\int [(\log (t) - \frac{1}{t}) + [\frac{1}{t} + \frac{1}{{(t)}^{2}}]] e^{t} d t = (\log (t) - \frac{1}{t}) e^{t} = x (\log (\log x) - \frac{1}{\log x}) + c$

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