What is the integral of e^{\sqrt[3]{x}} ?

The integral of e^{\sqrt[3]{x}} is -(x\Gamma(1/(1/3),-x^{1/3}))/(1/3 \sqrt[ 1/3 ]{-x^{1/3)}}+C

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integral of e^{\sqrt[3]{x}}

\int e^{3 x} d x

- \frac{x Γ ( \frac{1}{\frac{1}{3}} , - x ^{\frac{1}{3}} )}{\frac{1}{3} \frac{1}{3} - x ^{\frac{1}{3}}} + C

Solution steps

\int e^{3 x} d x

Use the nonelementary integral:

\int e^{x^{a}} d x = - \frac{x Γ ( \frac{1}{a} , - x ^{a} )}{a a - x ^{a}}

= - \frac{x Γ ( \frac{1}{\frac{1}{3}} , - x ^{\frac{1}{3}} )}{\frac{1}{3} \frac{1}{3} - x ^{\frac{1}{3}}}

Add a constant to the solution

= - \frac{x Γ ( \frac{1}{\frac{1}{3}} , - x ^{\frac{1}{3}} )}{\frac{1}{3} \frac{1}{3} - x ^{\frac{1}{3}}} + C

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What is the integral of e^{\sqrt[3]{x}} ?
The integral of e^{\sqrt[3]{x}} is -(x\Gamma(1/(1/3),-x^{1/3}))/(1/3 \sqrt[ 1/3 ]{-x^{1/3)}}+C