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

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

Sea $$$u=\frac{x}{5}$$$.

Entonces $$$du=\left(\frac{x}{5}\right)^{\prime }dx = \frac{dx}{5}$$$ (los pasos pueden verse »), y obtenemos que $$$dx = 5 du$$$.

Por lo tanto,

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

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=5$$$ y $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

$${\color{red}{\int{5 \sin{\left(u \right)} d u}}} = {\color{red}{\left(5 \int{\sin{\left(u \right)} d u}\right)}}$$

La integral del seno es $$$\int{\sin{\left(u \right)} d u} = - \cos{\left(u \right)}$$$:

$$5 {\color{red}{\int{\sin{\left(u \right)} d u}}} = 5 {\color{red}{\left(- \cos{\left(u \right)}\right)}}$$

Recordemos que $$$u=\frac{x}{5}$$$:

$$- 5 \cos{\left({\color{red}{u}} \right)} = - 5 \cos{\left({\color{red}{\left(\frac{x}{5}\right)}} \right)}$$

Por lo tanto,

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

Añade la constante de integración:

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

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