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

The integral of (ln(3x))/(\sqrt[3]{x)} is 3/2 x^{2/3}ln(3x)-9/4 x^{2/3}+C

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

\frac{3}{2} x^{\frac{2}{3}} ln (3 x) - \frac{9}{4} x^{\frac{2}{3}} + C

Solution steps

Apply Integration By Parts

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

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

\int \frac{3}{2 x ^{\frac{1}{3}}} d x = \frac{9}{4} x^{\frac{2}{3}}

= \frac{3}{2} x^{\frac{2}{3}} ln (3 x) - \frac{9}{4} x^{\frac{2}{3}}

Add a constant to the solution

= \frac{3}{2} x^{\frac{2}{3}} ln (3 x) - \frac{9}{4} x^{\frac{2}{3}} + C

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What is the integral of (ln(3x))/(\sqrt[3]{x)} ?
The integral of (ln(3x))/(\sqrt[3]{x)} is 3/2 x^{2/3}ln(3x)-9/4 x^{2/3}+C