Integrate the function: 1 x ( x ) ^m ,x > 0, m 1 - Vidyadip

Question

Integrate the function: $\frac{1}{{x{{\left( {\log x} \right)}^m}}},x > 0$, m $\neq$ 1

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Answer

Let $I = \int {\frac{1}{{x{{\left( {\log x} \right)}^m}}}} dx$
$= \int {\frac{{\frac{1}{x}dx}}{{{{\left( {\log x} \right)}^m}}}} dx$…(i)
Putting $\log x = t$
$ \Rightarrow \frac{1}{x} = \frac{{dt}}{{dx}}$
$\Rightarrow \frac{{dx}}{x} = dt$
$\therefore $ From eq. (i), $I = \int {\frac{{dt}}{{{t^m}}} = \int {{t^{ - m}}dt} } $
$ = \frac{{{t^{ - m + 1}}}}{{ - m + 1}} + c$
$= \frac{{{{\left( {\log x} \right)}^{1 - m}}}}{{1 - m}} + c$

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