How do I quickly spot that $\int_0^1 \frac{dx}{\sqrt{x(1-x^3)}}$ is convergent? I am really lost at figuring these problems out and need som help understanding the way of thinking. Thank you!
2026-04-09 07:24:39.1775719479
Trouble with improper integral convergence
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Quickly! Use the asymptotic comparison:
$$\frac1{\sqrt{x(1-x^3)}}\sim_0 \frac1{\sqrt x}$$ and the integral $$\int_0^{1/2}\frac{dx}{\sqrt x}$$ is convergent. Moreover, $$\frac1{\sqrt{x(1-x^3)}}\sim_1\frac1{\sqrt3\sqrt{1-x}}$$ and the integral $$\int_{1/2}^1\frac{dx}{\sqrt {1-x}}$$ is also convergent. Conclude.