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

The integral of sqrt((x^3))ln(\sqrt[3]{x}) is 1/3 (2/5 x^{5/2}ln(x)-4/25 x^{5/2})+C

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

\int (x^{3}) ln (3 x) d x

\frac{1}{3} (\frac{2}{5} x^{\frac{5}{2}} ln (x) - \frac{4}{25} x^{\frac{5}{2}}) + C

Solution steps

\int x^{3} ln (3 x) d x

Simplify

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

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

Take the constant out:

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

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

Apply Integration By Parts

: \frac{2}{5} x^{\frac{5}{2}} ln (x) - \int \frac{2}{5} x^{\frac{3}{2}} d x

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

\int \frac{2}{5} x^{\frac{3}{2}} d x = \frac{4}{25} x^{\frac{5}{2}}

= \frac{1}{3} (\frac{2}{5} x^{\frac{5}{2}} ln (x) - \frac{4}{25} x^{\frac{5}{2}})

Add a constant to the solution

= \frac{1}{3} (\frac{2}{5} x^{\frac{5}{2}} ln (x) - \frac{4}{25} x^{\frac{5}{2}}) + C

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What is the integral of sqrt((x^3))ln(\sqrt[3]{x}) ?
The integral of sqrt((x^3))ln(\sqrt[3]{x}) is 1/3 (2/5 x^{5/2}ln(x)-4/25 x^{5/2})+C