Integral of Error Function

Question

Is there a way to approximate this integral with a constant expressed in terms of $\delta$

$$\int_{0}^{1} e^{-\left(\frac{x^2}{2\delta^2 }\right)} dx$$

Thanks

Answer 1

Let $\frac{x}{\delta}=y$

Now your integral becomes

$$\delta \sqrt{2\pi}\int_{0}^{\frac{1}{\delta}}\frac{1}{\sqrt{2\pi}}e^{-\frac{y^2}{2}}dy=\delta \sqrt{2\pi}[\Phi(\frac{1}{\delta})-\Phi(0)]=\delta \sqrt{2\pi}[\Phi(\frac{1}{\delta})-\frac{1}{2}]$$

The values of $\Phi(x)$ are tabulated everywhere...it's the CDF of the Standard Gaussian

To approximate this integral a lot of methods are available. A pretty Statistical method is the "Importance Sampling"