Integral de $$$28 x z \ln\left(x^{2}\right)$$$ con respecto a $$$x$$$

Halla $$$\int 28 x z \ln\left(x^{2}\right)\, dx$$$.

La entrada se reescribe: $$$\int{28 x z \ln{\left(x^{2} \right)} d x}=\int{56 x z \ln{\left(x \right)} d x}$$$.

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=56 z$$$ y $$$f{\left(x \right)} = x \ln{\left(x \right)}$$$:

$${\color{red}{\int{56 x z \ln{\left(x \right)} d x}}} = {\color{red}{\left(56 z \int{x \ln{\left(x \right)} d x}\right)}}$$

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 »).

Entonces,

$$56 z {\color{red}{\int{x \ln{\left(x \right)} d x}}}=56 z {\color{red}{\left(\ln{\left(x \right)} \cdot \frac{x^{2}}{2}-\int{\frac{x^{2}}{2} \cdot \frac{1}{x} d x}\right)}}=56 z {\color{red}{\left(\frac{x^{2} \ln{\left(x \right)}}{2} - \int{\frac{x}{2} d x}\right)}}$$

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 $$$f{\left(x \right)} = x$$$:

$$56 z \left(\frac{x^{2} \ln{\left(x \right)}}{2} - {\color{red}{\int{\frac{x}{2} d x}}}\right) = 56 z \left(\frac{x^{2} \ln{\left(x \right)}}{2} - {\color{red}{\left(\frac{\int{x d x}}{2}\right)}}\right)$$

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

$$56 z \left(\frac{x^{2} \ln{\left(x \right)}}{2} - \frac{{\color{red}{\int{x d x}}}}{2}\right)=56 z \left(\frac{x^{2} \ln{\left(x \right)}}{2} - \frac{{\color{red}{\frac{x^{1 + 1}}{1 + 1}}}}{2}\right)=56 z \left(\frac{x^{2} \ln{\left(x \right)}}{2} - \frac{{\color{red}{\left(\frac{x^{2}}{2}\right)}}}{2}\right)$$

Por lo tanto,

$$\int{56 x z \ln{\left(x \right)} d x} = 56 z \left(\frac{x^{2} \ln{\left(x \right)}}{2} - \frac{x^{2}}{4}\right)$$

Simplificar:

$$\int{56 x z \ln{\left(x \right)} d x} = 14 x^{2} z \left(2 \ln{\left(x \right)} - 1\right)$$

Añade la constante de integración:

$$\int{56 x z \ln{\left(x \right)} d x} = 14 x^{2} z \left(2 \ln{\left(x \right)} - 1\right)+C$$

$$$\int 28 x z \ln\left(x^{2}\right)\, dx = 14 x^{2} z \left(2 \ln\left(x\right) - 1\right) + C$$$A