How to integrate $\displaystyle\int e^{\frac{x^3}{3}} \mathrm{d}x$

Question

While solving $y'=x^2-e^y$ I'm stuck on the last step that requires to evaluate this integral.

$$\displaystyle\int e^{\frac{x^3}{3}} \mathrm{d}x$$

I don't know how to approach it. I know that it will result in incomplete Gamma function along with some other stuff.

$\displaystyle\int_0^\infty e^{-x^{\large n}}dx~=~\Gamma\bigg(1+\dfrac1n\bigg)$ — Lucian, Nov 12 '14 at 01:59

score 2 · Accepted Answer · answered Nov 11 '14 at 18:30

2

$$\int e^{\frac{x^3}{3}} dx=-\int e^{-t} (-3t)^{-2/3}dt=(-3)^{-2/3}\Gamma\big(\frac{1}{3},t\big)=(-3)^{-2/3}\Gamma\big(\frac{1}{3},\frac{-x^3}{3}\big)$$ involving the Incomplete Gamma function.

answered Nov 11 '14 at 18:30

JJacquelin

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How to integrate $\displaystyle\int e^{\frac{x^3}{3}} \mathrm{d}x$

1 Answers1