# What is the antiderivative of e−x2e^{-x^2}

I was wondering what the antiderivative of $e^{-x^2}$ was, and when I wolfram alpha’d it I got

So, I of course didn’t know what this $\text{erf}$ was and I looked it up on wikipedia, where it was defined as:

To my mathematically illiterate mind, this is a bit too circular to understand. Why can’t we express $\int e^{-x^2} \textrm{d}x$ as a ‘normal function’? Also, what is the use of the error function?

When $$ff$$ is a continuous function on the interval $$[a,b][a,b]$$, we can find a function $$FF$$ defined on $$[a,b][a,b]$$ such that $$F′(x)=f(x)F'(x)=f(x)$$ for all $$x∈[a,b]x\in[a,b]$$. This is the “fundamental theorem of calculus”; just consider
$$F(x)=∫xaf(t)dt F(x)=\int_{a}^{x} f(t)\,dt$$
There are other functions with the same property, precisely those of the form $$F(x)+cF(x)+c$$ where $$cc$$ is a completely arbitrary constant.
Sometimes this function can be expressed with the so-called “elementary functions”, that is, polynomials, rational functions, exponential, logarithm, trigonometric functions and any algebraic combination thereof. Some (actually many) functions do not admit an antiderivative expressible in this form; it’s the case of $$e−x2e^{-x^2}$$ and it can be proved, although not easily.
Think of a simpler example: if all we have available as “elementary functions” are polynomials or, more generally, rational functions, the function $$1/x1/x$$ wouldn’t admit an “elementary antiderivative”, but it still would have one:
$$∫x11tdt \int_{1}^{x}\frac{1}{t}\,dt$$
Since this is a “new” function, we give it a name, precisely “$$log\log$$” and we have extended the tool set. The same happens with “$$erf\operatorname{erf}$$”, which has many uses in probability theory and statistics, being related to normal distributions.