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

The integral of \sqrt[3]{e^x} is 3e^{x/3}+C

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

3 e^{\frac{x}{3}} + C

Solution steps

Apply radical rule:

assuming

a \geq 0

= \int e^{\frac{x}{3}} d x

Apply u-substitution

= \int e^{u} \cdot 3 d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int e^{u} d u

Use the common integral:

\int e^{u} d u = e^{u}

= 3 e^{u}

Substitute back

u = \frac{x}{3}

= 3 e^{\frac{x}{3}}

Add a constant to the solution

= 3 e^{\frac{x}{3}} + C

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