I have to calculate if this integral is convergent for any "$a$":
$$ \int_2^\infty \frac{1}{x\log^a(x)} dx $$
(I made sure to not make any mistake when writing the integral from paper into this forums!)
I really don't what is meant by this..
2026-05-15 11:29:20.1778844560
Integral convergent calculating
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Changing variable to $t=\log x$, one has $$ \int_2^{+\infty}\frac{1}{x\log^{\alpha} x}\textrm{d}x=\int_{\log 2}^{+\infty}\frac{1}{t^{\alpha} }\textrm{d}t, $$ so the integral converges if $\alpha>1$, diverges if $\alpha\le 1$.