First slide

Methods of integration

Question

If $f_{n} (x) = \log \log \dots \log x$ (log is repeated n times)
then $\int {[x f_{1} (x) f_{2} (x) \dots f_{n} (x)]}^{- 1} d x$ is equal to

Easy

A

$f_{n + 1} (x) + C$
B

$\frac{f_{n + 1} (x)}{n + 1} + C$
C

$n f_{n} (x) + C$
D

none of these

Solution

Put $f_{n} (x) = t, \frac{d t}{d x} = \frac{1}{x f_{1} (x) \dots f_{n - 1} (x)}$
So $\int {[x f_{1} (x) \dots f_{n} (x)]}^{- 1} d x = \int \frac{d t}{t} = \log t + C$
$= f_{n + 1} (x) + C .$

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