How do you find the integral of #e^(x^2)#?

Question

Emily Ammerman · Accepted Answer

Use some kind of approximation method.  There is no nice, finitely expressible antiderivative.   (Other that to write:  #int e^(x^2) dx#,  of course.)

Charles Arocha · Answer

One symbolic way to do it is to use infinite series.  Since #e^{x}=1+x+x^{2}/{2!}+x^{3}/{3!}+\cdots=1+x+x^{2}/2+x^{3}/6+\cdots# (for all #x#), it follows that #e^{x^{2}}=1+x^{2}+x^{4}/2+x^{6}/6+\cdots# (for all #x#).   It is valid in this example to now integrate term-by-term (the result is true for all #x#): #\int e^{x^{2}} dx=\int (1+x^{2}+x^{4}/2+x^{6}/6+\cdots) dx# #=C+x+x^{3}/3+x^{5}/10+x^{7}/42+\cdots#. Alternatively, you can also give the antiderivative a name.  Wolfram Alpha writes the antiderivative whose graph goes through the origin as #\frac{\sqrt{\pi}}{2}\mbox{erfi}(x)#, where #\mbox{erfi}(x)# is called the "imaginary error function".

Scarlett Allison · Answer

undefined The integral of e^(x^2) with respect to x does not have an elementary antiderivative in terms of standard mathematical functions. However, it can be expressed in terms of a special function called the error function, denoted as erf(x). The integral of e^(x^2) with respect to x is typically denoted as ∫ e^(x^2) dx = √π * erf(x) + C, where C is the constant of integration.