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Hi could you help me with the following

$\int_0^\infty \frac{dx}{x^2 (\sin x)^{2/3}}$

How can I show that this integral is convergent? Thank you.

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    That's a definite, not an indefinite, integral.2012-09-09

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If you are interested in $\int_0^\infty \frac{dx}{x^2((\sin x)^2)^{1/3}},$ then you cannot show the integral is convergent, since it is in fact divergent. Informally, the function blows up too rapidly near $0$ for the integral to exist. If you want to do a formal comparison, do so for example with $\int_0^\infty \frac{dx}{x^2}$, which diverges. The problem is that $\int_{\epsilon}^1 \frac{dx}{x^2}$ blows up as $\epsilon$ approaches $0$ from the right.