What is the integral of \sqrt[3]{5^x} ?

The integral of \sqrt[3]{5^x} is 3/(ln(5))*5^{x/3}+C

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integral of \sqrt[3]{5^x}

\frac{3}{ln ( 5 )} \cdot 5^{\frac{x}{3}} + C

Solution steps

Apply u-substitution

= \int \frac{1}{ln ( 5 ) u ^{\frac{2}{3}}} d u

Take the constant out:

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

= \frac{1}{ln ( 5 )} \cdot \int \frac{1}{u ^{\frac{2}{3}}} d u

Apply the Power Rule

= \frac{1}{ln ( 5 )} \cdot 3 u^{\frac{1}{3}}

Substitute back

u = 5^{x}

= \frac{1}{ln ( 5 )} \cdot 3 (5^{x})^{\frac{1}{3}}

Simplify

\frac{1}{ln ( 5 )} \cdot 3 (5^{x})^{\frac{1}{3}} : \frac{3}{ln ( 5 )} \cdot 5^{\frac{x}{3}}

= \frac{3}{ln ( 5 )} \cdot 5^{\frac{x}{3}}

Add a constant to the solution

= \frac{3}{ln ( 5 )} \cdot 5^{\frac{x}{3}} + C

What is the integral of \sqrt[3]{5^x} ?
The integral of \sqrt[3]{5^x} is 3/(ln(5))*5^{x/3}+C