Integral of $$$\sqrt[3]{x} \ln\left(x\right)$$$

Find $$$\int \sqrt[3]{x} \ln\left(x\right)\, dx$$$.

For the integral $$$\int{\sqrt[3]{x} \ln{\left(x \right)} d x}$$$, use integration by parts $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Let $$$\operatorname{u}=\ln{\left(x \right)}$$$ and $$$\operatorname{dv}=\sqrt[3]{x} dx$$$.

Then $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (steps can be seen ») and $$$\operatorname{v}=\int{\sqrt[3]{x} d x}=\frac{3 x^{\frac{4}{3}}}{4}$$$ (steps can be seen »).

Thus,

$${\color{red}{\int{\sqrt[3]{x} \ln{\left(x \right)} d x}}}={\color{red}{\left(\ln{\left(x \right)} \cdot \frac{3 x^{\frac{4}{3}}}{4}-\int{\frac{3 x^{\frac{4}{3}}}{4} \cdot \frac{1}{x} d x}\right)}}={\color{red}{\left(\frac{3 x^{\frac{4}{3}} \ln{\left(x \right)}}{4} - \int{\frac{3 \sqrt[3]{x}}{4} d x}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{3}{4}$$$ and $$$f{\left(x \right)} = \sqrt[3]{x}$$$:

$$\frac{3 x^{\frac{4}{3}} \ln{\left(x \right)}}{4} - {\color{red}{\int{\frac{3 \sqrt[3]{x}}{4} d x}}} = \frac{3 x^{\frac{4}{3}} \ln{\left(x \right)}}{4} - {\color{red}{\left(\frac{3 \int{\sqrt[3]{x} d x}}{4}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=\frac{1}{3}$$$:

$$\frac{3 x^{\frac{4}{3}} \ln{\left(x \right)}}{4} - \frac{3 {\color{red}{\int{\sqrt[3]{x} d x}}}}{4}=\frac{3 x^{\frac{4}{3}} \ln{\left(x \right)}}{4} - \frac{3 {\color{red}{\int{x^{\frac{1}{3}} d x}}}}{4}=\frac{3 x^{\frac{4}{3}} \ln{\left(x \right)}}{4} - \frac{3 {\color{red}{\frac{x^{\frac{1}{3} + 1}}{\frac{1}{3} + 1}}}}{4}=\frac{3 x^{\frac{4}{3}} \ln{\left(x \right)}}{4} - \frac{3 {\color{red}{\left(\frac{3 x^{\frac{4}{3}}}{4}\right)}}}{4}$$

Therefore,

$$\int{\sqrt[3]{x} \ln{\left(x \right)} d x} = \frac{3 x^{\frac{4}{3}} \ln{\left(x \right)}}{4} - \frac{9 x^{\frac{4}{3}}}{16}$$

Simplify:

$$\int{\sqrt[3]{x} \ln{\left(x \right)} d x} = \frac{3 x^{\frac{4}{3}} \left(4 \ln{\left(x \right)} - 3\right)}{16}$$

Add the constant of integration:

$$\int{\sqrt[3]{x} \ln{\left(x \right)} d x} = \frac{3 x^{\frac{4}{3}} \left(4 \ln{\left(x \right)} - 3\right)}{16}+C$$

$$$\int \sqrt[3]{x} \ln\left(x\right)\, dx = \frac{3 x^{\frac{4}{3}} \left(4 \ln\left(x\right) - 3\right)}{16} + C$$$A