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

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

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

\int x ln (3 x) d x

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

Solution steps

\int x ln (3 x) d x

Simplify

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

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

Apply Integration By Parts

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

\int \frac{x}{2} d x = \frac{x ^{2}}{4}

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

Add a constant to the solution

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

t an g e n t f (x) = - \frac{1}{x}

d er i v a t i v e f (x) = 1 + x

\int \frac{\frac{π}{6}}{0} sin (x) cos^{3} (x) d x

in v erse l a pl a ce \frac{1}{s ^{2}}

\int - \frac{2000}{x ^{2}} d x

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