MHB Can Improper Integrals Help Solve This Inequality?

ozgunozgur
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This is my method, could you help me to continue?

 
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note ...

$\dfrac{x\sqrt{x}}{x^2-1} > \dfrac{1}{\sqrt{x}}$

and $\displaystyle \int_2^\infty \dfrac{dx}{\sqrt{x}}$ is divergent.
 

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