Integral de $$$\frac{k \sin{\left(\frac{x}{k} \right)}}{x}$$$ con respecto a $$$x$$$

Halla $$$\int \frac{k \sin{\left(\frac{x}{k} \right)}}{x}\, dx$$$.

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

$${\color{red}{\int{\frac{k \sin{\left(\frac{x}{k} \right)}}{x} d x}}} = {\color{red}{k \int{\frac{\sin{\left(\frac{x}{k} \right)}}{x} d x}}}$$

Sea $$$u=\frac{x}{k}$$$.

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

Por lo tanto,

$$k {\color{red}{\int{\frac{\sin{\left(\frac{x}{k} \right)}}{x} d x}}} = k {\color{red}{\int{\frac{\sin{\left(u \right)}}{u} d u}}}$$

Esta integral (Integral seno) no tiene una forma cerrada:

$$k {\color{red}{\int{\frac{\sin{\left(u \right)}}{u} d u}}} = k {\color{red}{\operatorname{Si}{\left(u \right)}}}$$

Recordemos que $$$u=\frac{x}{k}$$$:

$$k \operatorname{Si}{\left({\color{red}{u}} \right)} = k \operatorname{Si}{\left({\color{red}{\frac{x}{k}}} \right)}$$

Por lo tanto,

$$\int{\frac{k \sin{\left(\frac{x}{k} \right)}}{x} d x} = k \operatorname{Si}{\left(\frac{x}{k} \right)}$$

Añade la constante de integración:

$$\int{\frac{k \sin{\left(\frac{x}{k} \right)}}{x} d x} = k \operatorname{Si}{\left(\frac{x}{k} \right)}+C$$

$$$\int \frac{k \sin{\left(\frac{x}{k} \right)}}{x}\, dx = k \operatorname{Si}{\left(\frac{x}{k} \right)} + C$$$A