Integral de $$$\frac{\sin{\left(5 x \right)}}{2 \sin{\left(\frac{x_{0}}{5} \right)}}$$$ con respecto a $$$x$$$
Calculadora relacionada: Calculadora de integrales definidas e impropias
Tu entrada
Halla $$$\int \frac{\sin{\left(5 x \right)}}{2 \sin{\left(\frac{x_{0}}{5} \right)}}\, dx$$$.
Solución
Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=\frac{1}{2 \sin{\left(\frac{x_{0}}{5} \right)}}$$$ y $$$f{\left(x \right)} = \sin{\left(5 x \right)}$$$:
$${\color{red}{\int{\frac{\sin{\left(5 x \right)}}{2 \sin{\left(\frac{x_{0}}{5} \right)}} d x}}} = {\color{red}{\left(\frac{\int{\sin{\left(5 x \right)} d x}}{2 \sin{\left(\frac{x_{0}}{5} \right)}}\right)}}$$
Sea $$$u=5 x$$$.
Entonces $$$du=\left(5 x\right)^{\prime }dx = 5 dx$$$ (los pasos pueden verse »), y obtenemos que $$$dx = \frac{du}{5}$$$.
La integral se convierte en
$$\frac{{\color{red}{\int{\sin{\left(5 x \right)} d x}}}}{2 \sin{\left(\frac{x_{0}}{5} \right)}} = \frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{5} d u}}}}{2 \sin{\left(\frac{x_{0}}{5} \right)}}$$
Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{5}$$$ y $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:
$$\frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{5} d u}}}}{2 \sin{\left(\frac{x_{0}}{5} \right)}} = \frac{{\color{red}{\left(\frac{\int{\sin{\left(u \right)} d u}}{5}\right)}}}{2 \sin{\left(\frac{x_{0}}{5} \right)}}$$
La integral del seno es $$$\int{\sin{\left(u \right)} d u} = - \cos{\left(u \right)}$$$:
$$\frac{{\color{red}{\int{\sin{\left(u \right)} d u}}}}{10 \sin{\left(\frac{x_{0}}{5} \right)}} = \frac{{\color{red}{\left(- \cos{\left(u \right)}\right)}}}{10 \sin{\left(\frac{x_{0}}{5} \right)}}$$
Recordemos que $$$u=5 x$$$:
$$- \frac{\cos{\left({\color{red}{u}} \right)}}{10 \sin{\left(\frac{x_{0}}{5} \right)}} = - \frac{\cos{\left({\color{red}{\left(5 x\right)}} \right)}}{10 \sin{\left(\frac{x_{0}}{5} \right)}}$$
Por lo tanto,
$$\int{\frac{\sin{\left(5 x \right)}}{2 \sin{\left(\frac{x_{0}}{5} \right)}} d x} = - \frac{\cos{\left(5 x \right)}}{10 \sin{\left(\frac{x_{0}}{5} \right)}}$$
Añade la constante de integración:
$$\int{\frac{\sin{\left(5 x \right)}}{2 \sin{\left(\frac{x_{0}}{5} \right)}} d x} = - \frac{\cos{\left(5 x \right)}}{10 \sin{\left(\frac{x_{0}}{5} \right)}}+C$$
Respuesta
$$$\int \frac{\sin{\left(5 x \right)}}{2 \sin{\left(\frac{x_{0}}{5} \right)}}\, dx = - \frac{\cos{\left(5 x \right)}}{10 \sin{\left(\frac{x_{0}}{5} \right)}} + C$$$A