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Integral de $$$\sin{\left(\frac{x}{3} \right)}$$$
Calculadora relacionada: Calculadora de integrales definidas e impropias
Tu entrada
Halla $$$\int \sin{\left(\frac{x}{3} \right)}\, dx$$$.
Solución
Sea $$$u=\frac{x}{3}$$$.
Entonces $$$du=\left(\frac{x}{3}\right)^{\prime }dx = \frac{dx}{3}$$$ (los pasos pueden verse »), y obtenemos que $$$dx = 3 du$$$.
La integral puede reescribirse como
$${\color{red}{\int{\sin{\left(\frac{x}{3} \right)} d x}}} = {\color{red}{\int{3 \sin{\left(u \right)} d u}}}$$
Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=3$$$ y $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:
$${\color{red}{\int{3 \sin{\left(u \right)} d u}}} = {\color{red}{\left(3 \int{\sin{\left(u \right)} d u}\right)}}$$
La integral del seno es $$$\int{\sin{\left(u \right)} d u} = - \cos{\left(u \right)}$$$:
$$3 {\color{red}{\int{\sin{\left(u \right)} d u}}} = 3 {\color{red}{\left(- \cos{\left(u \right)}\right)}}$$
Recordemos que $$$u=\frac{x}{3}$$$:
$$- 3 \cos{\left({\color{red}{u}} \right)} = - 3 \cos{\left({\color{red}{\left(\frac{x}{3}\right)}} \right)}$$
Por lo tanto,
$$\int{\sin{\left(\frac{x}{3} \right)} d x} = - 3 \cos{\left(\frac{x}{3} \right)}$$
Añade la constante de integración:
$$\int{\sin{\left(\frac{x}{3} \right)} d x} = - 3 \cos{\left(\frac{x}{3} \right)}+C$$
Respuesta
$$$\int \sin{\left(\frac{x}{3} \right)}\, dx = - 3 \cos{\left(\frac{x}{3} \right)} + C$$$A