Integral de $$$x \ln\left(x^{2}\right)$$$

Halla $$$\int x \ln\left(x^{2}\right)\, dx$$$.

La entrada se reescribe: $$$\int{x \ln{\left(x^{2} \right)} d x}=\int{2 x \ln{\left(x \right)} d x}$$$.

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=2$$$ y $$$f{\left(x \right)} = x \ln{\left(x \right)}$$$:

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

Para la integral $$$\int{x \ln{\left(x \right)} d x}$$$, utiliza la integración por partes $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Sean $$$\operatorname{u}=\ln{\left(x \right)}$$$ y $$$\operatorname{dv}=x dx$$$.

Entonces $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (los pasos pueden verse ») y $$$\operatorname{v}=\int{x d x}=\frac{x^{2}}{2}$$$ (los pasos pueden verse »).

La integral puede reescribirse como

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

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=\frac{1}{2}$$$ y $$$f{\left(x \right)} = x$$$:

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

Aplica la regla de la potencia $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=1$$$:

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

Por lo tanto,

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

Simplificar:

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

Añade la constante de integración:

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

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