What is the integral of 2ln(\sqrt[3]{x}) ?

The integral of 2ln(\sqrt[3]{x}) is 2(1/3 xln(x)-1/3 x)+C

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integral of 2ln(\sqrt[3]{x})

\int 2 ln (3 x) d x

2 (\frac{1}{3} x ln (x) - \frac{1}{3} x) + C

Solution steps

\int 2 ln (3 x) d x

Take the constant out:

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

= 2 \cdot \int ln (3 x) d x

Apply Integration By Parts

: x ln (3 x) - \int \frac{1}{3} d x

= 2 (x ln (3 x) - \int \frac{1}{3} d x)

\int \frac{1}{3} d x = \frac{1}{3} x

= 2 (x ln (3 x) - \frac{1}{3} x)

Simplify

ln (3 x) : \frac{1}{3} ln (x)

= 2 (\frac{1}{3} x ln (x) - \frac{1}{3} x)

Add a constant to the solution

= 2 (\frac{1}{3} x ln (x) - \frac{1}{3} x) + C

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What is the integral of 2ln(\sqrt[3]{x}) ?
The integral of 2ln(\sqrt[3]{x}) is 2(1/3 xln(x)-1/3 x)+C