Integral de $$$\frac{\ln\left(x\right)}{5}$$$

Halla $$$\int \frac{\ln\left(x\right)}{5}\, dx$$$.

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=\frac{1}{5}$$$ y $$$f{\left(x \right)} = \ln{\left(x \right)}$$$:

$${\color{red}{\int{\frac{\ln{\left(x \right)}}{5} d x}}} = {\color{red}{\left(\frac{\int{\ln{\left(x \right)} d x}}{5}\right)}}$$

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

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

La integral se convierte en

$$\frac{{\color{red}{\int{\ln{\left(x \right)} d x}}}}{5}=\frac{{\color{red}{\left(\ln{\left(x \right)} \cdot x-\int{x \cdot \frac{1}{x} d x}\right)}}}{5}=\frac{{\color{red}{\left(x \ln{\left(x \right)} - \int{1 d x}\right)}}}{5}$$

Aplica la regla de la constante $$$\int c\, dx = c x$$$ con $$$c=1$$$:

$$\frac{x \ln{\left(x \right)}}{5} - \frac{{\color{red}{\int{1 d x}}}}{5} = \frac{x \ln{\left(x \right)}}{5} - \frac{{\color{red}{x}}}{5}$$

Por lo tanto,

$$\int{\frac{\ln{\left(x \right)}}{5} d x} = \frac{x \ln{\left(x \right)}}{5} - \frac{x}{5}$$

Simplificar:

$$\int{\frac{\ln{\left(x \right)}}{5} d x} = \frac{x \left(\ln{\left(x \right)} - 1\right)}{5}$$

Añade la constante de integración:

$$\int{\frac{\ln{\left(x \right)}}{5} d x} = \frac{x \left(\ln{\left(x \right)} - 1\right)}{5}+C$$

$$$\int \frac{\ln\left(x\right)}{5}\, dx = \frac{x \left(\ln\left(x\right) - 1\right)}{5} + C$$$A