Integral de $$$\ln\left(x_{0}\right)$$$

Halla $$$\int \ln\left(x_{0}\right)\, dx_{0}$$$.

Para la integral $$$\int{\ln{\left(x_{0} \right)} d x_{0}}$$$, utiliza la integración por partes $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Sean $$$\operatorname{u}=\ln{\left(x_{0} \right)}$$$ y $$$\operatorname{dv}=dx_{0}$$$.

Entonces $$$\operatorname{du}=\left(\ln{\left(x_{0} \right)}\right)^{\prime }dx_{0}=\frac{dx_{0}}{x_{0}}$$$ (los pasos pueden verse ») y $$$\operatorname{v}=\int{1 d x_{0}}=x_{0}$$$ (los pasos pueden verse »).

Entonces,

$${\color{red}{\int{\ln{\left(x_{0} \right)} d x_{0}}}}={\color{red}{\left(\ln{\left(x_{0} \right)} \cdot x_{0}-\int{x_{0} \cdot \frac{1}{x_{0}} d x_{0}}\right)}}={\color{red}{\left(x_{0} \ln{\left(x_{0} \right)} - \int{1 d x_{0}}\right)}}$$

Aplica la regla de la constante $$$\int c\, dx_{0} = c x_{0}$$$ con $$$c=1$$$:

$$x_{0} \ln{\left(x_{0} \right)} - {\color{red}{\int{1 d x_{0}}}} = x_{0} \ln{\left(x_{0} \right)} - {\color{red}{x_{0}}}$$

Por lo tanto,

$$\int{\ln{\left(x_{0} \right)} d x_{0}} = x_{0} \ln{\left(x_{0} \right)} - x_{0}$$

Simplificar:

$$\int{\ln{\left(x_{0} \right)} d x_{0}} = x_{0} \left(\ln{\left(x_{0} \right)} - 1\right)$$

Añade la constante de integración:

$$\int{\ln{\left(x_{0} \right)} d x_{0}} = x_{0} \left(\ln{\left(x_{0} \right)} - 1\right)+C$$

$$$\int \ln\left(x_{0}\right)\, dx_{0} = x_{0} \left(\ln\left(x_{0}\right) - 1\right) + C$$$A