How to integrate $\int_{0}^{x} e^{-t^2} dt$?

Question

I know how to derive $\int_{0}^{\infty} e^{-t^2} dt = \pi/2$ using polar coordinates, but what can I do if the upper limit is not infinity?

Answer 1

The integral you mentioned is continuous, and therefore Riemann integrable on $[0,x] $.

However, its antiderivative cannot be expressed in terms of elementary functions, which usually include polynomials, exponentials, logarithms, trigonometric functions, etc. Indeed, its antiderivative is a transformation of the Gauss error function:

$$\int_0^x e^{-t^2}\,dt=\frac{\sqrt{\pi}}{2} \text{ erf } x$$