Earlier today I stumbled across this when I was doing some practice questions for a physics course: $$\int e^{-t^2} \space \, \mathrm dt $$
To expand, the limits of integration were something like $1$ and $4$ (it was just a velocity function that needed to be integrated to find distance - it was not a known integral like $\int_0^\infty e^{-t^2} \space\, \mathrm dt$.)
Based on Wolfram|Alpha, it appears it cannot be expressed in elementary terms (i.e. it involves the error function.) Note that the questions involved the use of a calculator, so I was able to integrate the function using a CAS with ease, but I am wondering how to do it by hand. Thus, I was wondering if there was possibly a way to evaluate the integral using elementary methods from a calculus one or two course (read: no complex analysis). I thought there may perhaps be an elementary solution (I don't know what kind of algorithm Wolfram uses to evaluate integrals - I have seen them evaluate easy integrals in a lot of steps before.)
Answer
The error function is defined as
$$\operatorname{erf}(x)=\frac 2 {\sqrt \pi}\int_{0}^x e^{-t^2}dt$$
It is not an elementary function. Since from the definition it is immediate (FTCI) that
$$\operatorname{erf}'(x)=\frac 2 {\sqrt \pi}e^{-x^2}$$
the primitive of $e^{-x^2}$ is expressible as
$$\int e^{-x^2} dx =\frac{\sqrt \pi}{2}\operatorname{erf}(x)-C $$
since any two primitives of a function $f$ differ by a constant (FTCII)
As a consequence your primitive can't be expressed in terms of elementary functions.
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