Finding $F(x)$ so that $F'(x) = e^{-x^2}$

Question

Question: Let $f(x) = e^{-x^2}$. Find a formula for a function $F$ so that $F'(x) = f(x)$.

My Thoughts:

We have just proven the fundamental theorem of calculus, part II of which states:

Let $f$ be a continuous function on a finite interval $[a,b]$. Define $$F(x) := \int_a^xf(t)\, dt$$ Then $F \in C^1[a,b]$ and $F'(x) = f(x)$.

Taking advantage of this example in the theorem, and knowing from previous experience that $f(x) = e^{-x^2}$ has no elementary antiderivatives, should I set $$F(x) := \int_{-\infty}^x e^{-x^2}\,dx$$ and consider this $F(x)$ a suitable answer?

Edit: After the helpful comment and two answers, I see that $$ F(x) := \int_0^x e^{-t^2}\,dt$$ is an appropriate response to this question. Thank you.

Answer 1

This integral has no elementary antiderivative as you mention. However, we can set the lower limit to a finite constant, $a$ and get

$$F(x, a)=G(x)-G(a)=\int_a^x \exp(-t^2)\, dt$$

where

$$G(t)=\int \exp(-t^2)\, dt$$

so that

$$\frac{d}{dx}F(x, a)=\frac{d}{dx} G(x)=f(x)$$

If you would like, this function can be written in terms of the non-elementary error function if $a=0$:

$$F(x, 0)=\frac{\sqrt{\pi}}{2}\operatorname{erf}(x)$$

Answer 2

I would write it as $F(x) = \int_{-\infty}^x e^{-t^2}\,dt$ (distinguishing the free variable $x$ from the "dummy" integration variable $t$), but other than that, exactly.

Answer 3

Let $F(x) = \sum_{n\geq 0} (-1)^nx^{2n+1}/(2n+1)n!$.

Then $F'(x) = \sum_{n\geq 0} (-x^2)^n/n! = e^{-x^2}$.