Calculating the error function

Question

If I want to calculate the following integral in terms of the Error function, is this correct?

$$\frac{1}{\sqrt{2\pi}}\int_{f(x)}^{-\infty}e^{-p^2}\mathrm{d}p = \mathrm{Erf}(-\infty) - \mathrm{Erf}(f(x))$$

Answer 1

You can check that $$\frac{1}{\sqrt{2\pi}}\int_{f(x)}^{\infty}e^{-t^2}dt = \frac{1}{2\sqrt{2}}\big(1-\mathrm{erf}(f(x))\big).$$

Hint: Use the definition of $\mathrm{erf}(x)$:

$$\mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^2}dt$$ and the identity: $$\int_{-\infty}^{\infty}e^{-t^2}dt = \sqrt{\pi}.$$