<s>[INST] Problem on improper integrals I'm having some trouble evaluating the following integrals:
$$
I=\int_{-\infty}^0\frac{dx}{\sqrt{1-x^3}},\quad 
J=\int_0^1\frac{dx}{\sqrt{1-x^3}}.
$$
Any help will be appreciated.
 [/INST] The $J$ integral is reduced to the Euler integral of the first kind using $u$-substitution $u=x^3$:
$$
  J = \frac{1}{3} \int_0^1 \left(1-u\right)^{1/2-1} u^{1/3-1} \mathrm{d}u = \frac{1}{3} \operatorname{Beta}\left(\frac{1}{3}, \frac{1}{2} \right) 
$$
The $I$ integral can be reduced to the beta integral using $x^3 = \frac{u}{1-u}$, i.e. $u=\frac{x^3}{1+x^3}$:
$$
  I = \int_0^\infty \frac{\mathrm{d}x}{\sqrt{1+x^3}} =\frac{1}{3} \int_0^1 u^{1/3-1} \left(1-u\right)^{1/6-1} \mathrm{d}u = \frac{1}{3} \operatorname{Beta}\left(\frac{1}{3}, \frac{1}{6} \right) 
$$
</s>