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

Find $$$\int \frac{x \ln\left(x\right)}{2}\, dx$$$.

The input is rewritten: $$$\int{x \ln{\left(\sqrt{x} \right)} d x}=\int{\frac{x \ln{\left(x \right)}}{2} d x}$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{2}$$$ and $$$f{\left(x \right)} = x \ln{\left(x \right)}$$$:

$${\color{red}{\int{\frac{x \ln{\left(x \right)}}{2} d x}}} = {\color{red}{\left(\frac{\int{x \ln{\left(x \right)} d x}}{2}\right)}}$$

For the integral $$$\int{x \ln{\left(x \right)} d x}$$$, 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 $$$\operatorname{dv}=x dx$$$.

Then $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (steps can be seen ») and $$$\operatorname{v}=\int{x d x}=\frac{x^{2}}{2}$$$ (steps can be seen »).

Thus,

$$\frac{{\color{red}{\int{x \ln{\left(x \right)} d x}}}}{2}=\frac{{\color{red}{\left(\ln{\left(x \right)} \cdot \frac{x^{2}}{2}-\int{\frac{x^{2}}{2} \cdot \frac{1}{x} d x}\right)}}}{2}=\frac{{\color{red}{\left(\frac{x^{2} \ln{\left(x \right)}}{2} - \int{\frac{x}{2} d x}\right)}}}{2}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{2}$$$ and $$$f{\left(x \right)} = x$$$:

$$\frac{x^{2} \ln{\left(x \right)}}{4} - \frac{{\color{red}{\int{\frac{x}{2} d x}}}}{2} = \frac{x^{2} \ln{\left(x \right)}}{4} - \frac{{\color{red}{\left(\frac{\int{x d x}}{2}\right)}}}{2}$$

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

$$\frac{x^{2} \ln{\left(x \right)}}{4} - \frac{{\color{red}{\int{x d x}}}}{4}=\frac{x^{2} \ln{\left(x \right)}}{4} - \frac{{\color{red}{\frac{x^{1 + 1}}{1 + 1}}}}{4}=\frac{x^{2} \ln{\left(x \right)}}{4} - \frac{{\color{red}{\left(\frac{x^{2}}{2}\right)}}}{4}$$

Therefore,

$$\int{\frac{x \ln{\left(x \right)}}{2} d x} = \frac{x^{2} \ln{\left(x \right)}}{4} - \frac{x^{2}}{8}$$

Simplify:

$$\int{\frac{x \ln{\left(x \right)}}{2} d x} = \frac{x^{2} \left(2 \ln{\left(x \right)} - 1\right)}{8}$$

Add the constant of integration:

$$\int{\frac{x \ln{\left(x \right)}}{2} d x} = \frac{x^{2} \left(2 \ln{\left(x \right)} - 1\right)}{8}+C$$

$$$\int \frac{x \ln\left(x\right)}{2}\, dx = \frac{x^{2} \left(2 \ln\left(x\right) - 1\right)}{8} + C$$$A