An approximation of an integral Is there any good way to approximate following integral?
$$\int_0^{0.5}\frac{x^2}{\sqrt{2\pi}\sigma}\cdot \exp\left(-\frac{(x^2-\mu)^2}{2\sigma^2}\right)\mathrm dx$$
$\mu$ is between $0$ and $0.25$, the problem is in $\sigma$ which is always positive, but it can be arbitrarily small.
I was trying to expand it using Taylor series, but terms looks more or less this $\pm a_n\cdot\frac{x^{2n+3}}{\sigma^{2n}}$ and that can be arbitrarily large, so the error is significant.

Not an answer, but might still be helpful. Using the variable substitution that Ross mentions in his answer, we can treat a simpler case $\mu=0$ more easily. 
For the following integral, Wolfram Alpha tells us that (I hope I did not make a transcription error here):
$$\int_0^\infty \sqrt{y}e^{-y^2/(2\sigma^2)}dy = \frac{\sigma^{3/2}\Gamma(3/4)}{2^{1/4}},$$
But your problem goes from $0$ to $0.25$, so some approximations are needed.
Link-text: http://www.wolframalpha.com/input/?i=Integrate[Sqrt[x]+Exp[-x^2%2F%282+s^2%29]%2C{x%2C0%2Cinf}]
