Integral de $$$\cos{\left(8 x \right)}$$$

Halla $$$\int \cos{\left(8 x \right)}\, dx$$$.

Sea $$$u=8 x$$$.

Entonces $$$du=\left(8 x\right)^{\prime }dx = 8 dx$$$ (los pasos pueden verse »), y obtenemos que $$$dx = \frac{du}{8}$$$.

Entonces,

$${\color{red}{\int{\cos{\left(8 x \right)} d x}}} = {\color{red}{\int{\frac{\cos{\left(u \right)}}{8} d u}}}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{8}$$$ y $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\cos{\left(u \right)}}{8} d u}}} = {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{8}\right)}}$$

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

$$\frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{8} = \frac{{\color{red}{\sin{\left(u \right)}}}}{8}$$

Recordemos que $$$u=8 x$$$:

$$\frac{\sin{\left({\color{red}{u}} \right)}}{8} = \frac{\sin{\left({\color{red}{\left(8 x\right)}} \right)}}{8}$$

Por lo tanto,

$$\int{\cos{\left(8 x \right)} d x} = \frac{\sin{\left(8 x \right)}}{8}$$

Añade la constante de integración:

$$\int{\cos{\left(8 x \right)} d x} = \frac{\sin{\left(8 x \right)}}{8}+C$$

$$$\int \cos{\left(8 x \right)}\, dx = \frac{\sin{\left(8 x \right)}}{8} + C$$$A