convergence of $\int_{0}^{1}\frac{1}{\sqrt[3]{(1-x^2)^5}}\, dx$

Question

I would like to prove that this integral is divergent. Thanks for any suggestions.

$$\int_{0}^{1}\frac{1}{\sqrt[3]{(1-x^2)^5}}\, dx$$

Answer 1

Hint. A potential issue is as $x \to 1^-$, in this case we have $$ \frac{1}{\sqrt[3]{(1-x^2)^5}} \sim \frac{1}{2^{5/3}\cdot(1-x)^{5/3}} $$ the latter integrand is divergent over $[\varepsilon,1)$, $0<\varepsilon<1$, since $5/3>1$.