# How do you find the definite integral of #(dx)/ (sqrtxsin(sqrtx)# from #[pi^2, 4pi^2]#?

This integral does not converge.

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To find the definite integral of (\frac{dx}{\sqrt{x}\sin(\sqrt{x})}) from ([{\pi}^2, 4{\pi}^2]), you can use the substitution method. Let (u = \sqrt{x}), then (du = \frac{1}{2\sqrt{x}}dx). Rearrange to express (dx) in terms of (du). Substituting these into the integral, it becomes (\int \frac{2}{\sin(u)} du), which simplifies to (-2\ln|\csc(u) + \cot(u)| + C). Finally, replace (u) with (\sqrt{x}) and evaluate the integral from (\pi^2) to (4\pi^2) to get the definite integral value.

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