Does $ \int _1^{\infty }\frac{\sinh (a \log (x))}{\sqrt{x}} $ converge or diverge?

definite-integrals integration

Well we have

$$\sinh(a\log x) = \frac{x^a-x^{-a}}{2}$$

Thus the integral reduces to

$$\int_1^\infty \frac{x^a-x^{-a}}{2\sqrt{x}}dx$$

and this integral diverges for any $a\neq 0$ since would need simultaneously that $\frac{1}{2} < a < - \frac{1}{2}$

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