By the general of integration can we find the integration of #e^(x^2)# ?
1 Answer
# int_0^x \ e^(t^2) \ dt = sqrt(pi)/2 \ erfi(x)#
Explanation:
In theory any integral can be evaluated (either numerically, or using a powers series), However not every integral has an associated anti-derivative defined in terms of the well known elementary functions that all scientists know and love.
The integral:
# int \ e^(x^2) \ dx #
Falls into this category.
That is, there is no elementary function that we can differentiate to get the function
# erf(x) := 2/sqrt(pi) \ int_0^x \ e^(-t^2) \ dt #
Along with the Imaginary error function:
# erfi(x) := i \ erf(ix) = 2/sqrt(pi) \ int_0^x \ e^(t^2) \ dt #
And so the solution the the given integral is:
# int_0^x \ e^(t^2) \ dt = sqrt(pi)/2 \ erfi(x)#