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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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1  
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. –  André Nicolas Jan 5 '13 at 4:40

5 Answers 5

up vote 4 down vote accepted

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 $$\int e^{x^3}dx=\ x\ 1F1(\frac{1}{3};\frac{4}{3};x^3)+c$$

where 1F1 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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+1 nice, how did you come by this? –  Arjang May 20 '13 at 12:41
    
Really? Doesn't it diverge for $n>0$? –  Javier Badia May 20 '13 at 18:26
1  
Unless there's a minus sign in the exponential, the statement is false. –  johnny May 20 '13 at 18:31
    
@johnny: thanks, of course there should be a minus sign in the exponential. –  Eckhard May 20 '13 at 20:00

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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This answer seems correct for $x<0$. –  GEdgar May 20 '13 at 20:55

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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