Question about an integrable singularity

You can think of integrable singularity as a singularity of integrand that will disappear under suitable change of variables. Consider for example:

$$ \int_{0}^1 \frac{1}{\sqrt{1-x} } \mathrm{d} x $$ Then for $1-x = y^2$ and $-\mathrm{d}x = 2 y \, \mathrm{d} y$:

$$ \int_{0}^1 \frac{1}{\sqrt{1-x} } \mathrm{d} x = \int_{0}^1 \frac{1}{\sqrt{y^2} } 2 y \, \mathrm{d} y = 2 $$