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

Question

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 $\int\frac{e^t}{3\sqrt[3]{t^2}}dt$.

Answer 1

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).

Answer 2

The integral cannot be evaluated. We have to use power series of exponent and then integral term by term. $$e^{x}=\sum_{n \geq 0}{\frac{x^n}{n!}}$$

Answer 3

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