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How to evaluate the integral $$\int e^{x^3}dx \quad ?$$

I've tried to set $t=x^3$, but it seems to be a blind alley; I don't know what to do with $\int\frac{e^t}{3\sqrt[3]{t^2}}dt$.

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    $\begingroup$ It is very common for an elementary function not to have an elementary antiderivative. Proving this is the case for a particular function can be difficult. Your function $e^{x^3}$ happens to be one for which the standard method for showing "impossibility," which dates back in principle to Liouville, works reasonably smoothly. Many non-elementary "special functions" have been devised such that useful integrals can be expressed in terms of these special functions. I would guess that Maple, or Mathematica, even Wolfram Alpha, can produce an answer in terms of some special function. $\endgroup$ Jan 5, 2013 at 4:40

6 Answers 6

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The antiderivative of $e^{x^3}$ cannot be expressed in terms of elementary functions. We can, however, express it using power series. Since $$ e^x = \sum_{n \geq 0} \frac{x^n}{n!}, $$ $$ e^{x^3} = \sum_{n \geq 0} \frac{(x^3)^n}{n!} = \sum_{n \geq 0} \frac{x^{3n}}{n!}.$$

You can integrate term by term to find a series representation of the antiderivative (which converges on the entire complex plane, since $e^{x^3}$ is an entire function).

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$$\int e^{x^3}dx=\int \sum_{n=0}^{\infty }\frac{x^{3n}}{n!}dx$$

$$\int \sum_{n=0}^{\infty }\frac{x^{3n}}{n!}dx=\sum_{n=0}^{\infty }\frac{x^{3n+1}}{(3n+1)(n!)}+c$$

$$\frac{1}{3n+1}=\frac{(\frac{1}{3})^{(n)}}{(\frac{4}{3})^{(n)}}$$

$$\sum_{n=0}^{\infty }\frac{x^{3n+1}}{(3n+1)(n!)}+c=x\sum_{n=0}^{\infty }\frac{(\frac{1}{3})^{(n)}(x^3)^n}{(\frac{4}{3})^{(n)}(n!)}+c$$

$$x\sum_{n=0}^{\infty }\frac{(\frac{1}{3})^{(n)}(x^3)^n}{(\frac{4}{3})^{(n)}(n!)}+c=\ x\ 1F1(\frac{1}{3};\frac{4}{3};x^3)+c$$

so $$\int e^{x^3}dx=\ x\ {}_1F_1(\frac{1}{3};\frac{4}{3};x^3)+c$$

where ${}_1F_1$ is Hypergeometric Function of the First Kind

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Interestingly, the definite integral $$ \int_0^\infty e^{-x^n}dx $$ can be evaluated for any $n>0$, and is equal to $\Gamma((n+1)/n)$.

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    $\begingroup$ +1 nice, how did you come by this? $\endgroup$
    – jimjim
    May 20, 2013 at 12:41
  • $\begingroup$ Really? Doesn't it diverge for $n>0$? $\endgroup$
    – Javier
    May 20, 2013 at 18:26
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    $\begingroup$ Unless there's a minus sign in the exponential, the statement is false. $\endgroup$
    – johnny
    May 20, 2013 at 18:31
  • $\begingroup$ @johnny: thanks, of course there should be a minus sign in the exponential. $\endgroup$
    – Eckhard
    May 20, 2013 at 20:00
  • $\begingroup$ @Arjang you can prove it for example with the substitution $t=x^n$ $\endgroup$
    – glS
    May 5, 2015 at 11:05
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another try you can solve it with Gamma function

$$\int e^{x^3}dx=\frac{-1}{3}\int e^{-t}t^{\frac{1}{3}-1}dt$$

$$\frac{-1}{3}\int e^{-t}t^{\frac{1}{3}-1}dt=\frac{-1}{3}\int_{0}^{t}e^{-t}t^{\frac{1}{3}-1}dt+c$$

$$\frac{-1}{3}\int_{0}^{t}e^{-t}t^{\frac{1}{3}-1}dt+c=\frac{1}{3}(\int_{0}^{\infty }e^{-t}t^{\frac{1}{3}-1}dt-\int_{t}^{\infty }e^{-t}t^{\frac{1}{3}-1}dt)+c$$

$$\frac{-1}{3}(\int_{0}^{\infty }e^{-t}t^{\frac{1}{3}-1}dt-\int_{t}^{\infty }e^{-t}t^{\frac{1}{3}-1}dt)+c=\frac{1}{3}\Gamma (\frac{1}{3},t)+d$$

$$\frac{1}{3}\Gamma (\frac{1}{3},t)+d=\frac{1}{3}\Gamma (\frac{1}{3},-x^3)+d$$

so

$$\int e^{x^3}dx=\frac{1}{3}\Gamma (\frac{1}{3},-x^3)+d$$

where d and c are constant

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  • $\begingroup$ This answer seems correct for $x<0$. $\endgroup$
    – GEdgar
    May 20, 2013 at 20:55
  • $\begingroup$ @GEdgar Lack of convergence there. You'll need the lower gamma function or infinite constants of integration. $\endgroup$ Feb 1, 2017 at 0:48
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The integral cannot be evaluated. We have to use power series of exponent and then integral term by term. $$e^{x}=\sum_{n \geq 0}{\frac{x^n}{n!}}$$

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I'd like to give step by step solution to @Eckhard 's answer .

Substitude $y=x^n$ :

$\int_0^\infty e^{-x^n}dx = \frac1n\int_0^\infty e^{-y}y^{1/n-1}dy = \frac1n \Gamma(\frac 1n) = \Gamma(\frac 1n+1) = {\frac 1n}! $

$x^n=y \Rightarrow nx^{n-1}dx=dy \Rightarrow dx=\frac1nx^{1-n}dy \Rightarrow dx=\frac 1n y^{\frac 1n -1}dy$

The last integral is taken due to the main definition of gamma function .

Where :

$\Gamma(z) = \int_o^\infty e^{-x}x^{z-1}dx$

-Hope it was helpful.

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