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Quadratic substitution for a square-root endpoint singularity

If

f (x) = x^{- 1/2} g (x)

with

g

smooth near zero, the substitution

x = s^{2}

gives

\int_{0}^{1} f (x) d x = \int_{0}^{1} 2 g (s^{2}) d s .

The transformed integrand is smooth, so an ordinary second-order midpoint rule recovers its usual convergence rate.

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