Question:
Write a value of $\int \frac{1}{x(\log x)^{n}} d x$
Solution:
Let, $\log x=t$
Differentiating both sides with respect to $x$
$\frac{d t}{d x}=\frac{1}{x}$
$\Rightarrow d t=\frac{1}{x} d x$
$y=\int \frac{1}{t^{n}} d t$
Use formula $\int \frac{1}{t^{n}} d t=\frac{t^{-n+1}}{-n+1}$
$y=\frac{t^{-n+1}}{-n+1}+c$
Again, put $t=\log x$
$y=\frac{(\log x)^{-n+1}}{-n+1}+c$
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