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Can you please provide a step by step solution for next integral. I don't have any idea of how this can be solved $\displaystyle\int e^{x^2}\,dx$.

Jacob
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    Pretty sure your function has no antiderivative since it is not elementary. – Lemon Aug 29 '14 at 05:16
  • The function $e^{(x^2)}$ does not have an antiderivative expressible in terms of the usual elementary functions. If you mean $(e^x)^2$, that's easy, for then the function is $e^{2x}$. – André Nicolas Aug 29 '14 at 05:17
  • Perhaps of similar interest: http://math.stackexchange.com/questions/530611/derivative-of-xx2 – Display Name Aug 29 '14 at 05:33
  • See this question – JimmyK4542 Aug 29 '14 at 05:34
  • This may be what you are looking for...http://en.m.wikipedia.org/wiki/Error_function – ClassicStyle Aug 29 '14 at 05:35
  • Just to be clear, are you looking for the antiderivative of $e^{x^2}$ or are you looking to evaluate $\int_a^b e^{x^2},dx$ for specific values of $a$ and $b$? – JimmyK4542 Aug 29 '14 at 05:37
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    In a 2nd term calc course, you probably were given the task of finding the definie integral $\int_{-\infty}^{\infty}e^{x^2}dx$. That is easy using the trick of multiplying by itself with a dummy argument of $y$ instead of $x$, then transforming to polar coordinates, where you get $2\pi \int_{0}^{\infty}re^{r^2}dr$ – Mark Fischler Aug 29 '14 at 06:07

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As said in comments $e^{x^2}$ does not have an antiderivative expressible in terms of elementary functions.

In fact, $$\displaystyle\int e^{x^2}\,dx=\frac{\sqrt{\pi }}{2} \text{erfi}(x)$$ in which appears the imaginary error function defined by $$\text{erfi}(x)=\frac{\text{erf(ix)}}{i}$$ with $$\text{erf(z)}=\frac{2}{\sqrt{\pi }}\int_0^ze^{-t^2}dt$$ which is a definition.

Claude Leibovici
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