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I am stuck in this problem of integrating $e^{x^2}$. I was solving the linear differential equation of second order for damped oscillations in which i got this to solve

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    $\begingroup$ $e^{x^2}$ or $({e^x})^2$? $\endgroup$
    – Zelos Malum
    Mar 10, 2016 at 7:37
  • $\begingroup$ If you are integrating over the reals, it becomes $-i\sqrt{\pi}$ $\endgroup$
    – Bobson Dugnutt
    Mar 10, 2016 at 7:57

1 Answer 1

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The integral of the function $f(x)= e^{x^2}$ cannot be expressed in terms of elementary functions.

The integral can be given in terms of the imaginary error function, $\text{erfi}(x)$, which is defined as: $$\text{erfi}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^{x} e^{t^2} dt $$

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