Integral de $$$\ln\left(\frac{x}{x_{0}}\right)$$$ con respecto a $$$x$$$

Halla $$$\int \ln\left(\frac{x}{x_{0}}\right)\, dx$$$.

Sea $$$u=\frac{x}{x_{0}}$$$.

Entonces $$$du=\left(\frac{x}{x_{0}}\right)^{\prime }dx = \frac{dx}{x_{0}}$$$ (los pasos pueden verse »), y obtenemos que $$$dx = x_{0} du$$$.

Por lo tanto,

$${\color{red}{\int{\ln{\left(\frac{x}{x_{0}} \right)} d x}}} = {\color{red}{\int{x_{0} \ln{\left(u \right)} d u}}}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=x_{0}$$$ y $$$f{\left(u \right)} = \ln{\left(u \right)}$$$:

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

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

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

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

Por lo tanto,

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

Aplica la regla de la constante $$$\int c\, du = c u$$$ con $$$c=1$$$:

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

Recordemos que $$$u=\frac{x}{x_{0}}$$$:

$$x_{0} \left(- {\color{red}{u}} + {\color{red}{u}} \ln{\left({\color{red}{u}} \right)}\right) = x_{0} \left(- {\color{red}{\frac{x}{x_{0}}}} + {\color{red}{\frac{x}{x_{0}}}} \ln{\left({\color{red}{\frac{x}{x_{0}}}} \right)}\right)$$

Por lo tanto,

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

Simplificar:

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

Añade la constante de integración:

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

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