First slide

Methods of integration

Question

$If \int \frac{\log_{e} (\log_{e} x)}{x (\log_{e} x)} d x = f (x) and f (e) = 0 then f (e^{e}) = \dots \dots$

Moderate

A

0
B

1
C

$\frac{1}{2}$
D

$\frac{1}{4}$

Solution

$\begin{array}{l} Put \log_{e} (\log_{e} x) = t then \frac{1}{x (\log_{e} x)} d x = d t \\ f (x) = \int t d t = \frac{t^{2}}{2} + c = \frac{1}{2} {[\log_{e} (\log_{e} x)]}^{2} + c \\ f (e) = 0 \Rightarrow c = 0 \\ ∴ f (e^{e}) = \frac{1}{2} \end{array}$

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