Integral de $$$1 - \frac{\cos{\left(2 x \right)}}{2}$$$

Halla $$$\int \left(1 - \frac{\cos{\left(2 x \right)}}{2}\right)\, dx$$$.

Integra término a término:

$${\color{red}{\int{\left(1 - \frac{\cos{\left(2 x \right)}}{2}\right)d x}}} = {\color{red}{\left(\int{1 d x} - \int{\frac{\cos{\left(2 x \right)}}{2} d x}\right)}}$$

Aplica la regla de la constante $$$\int c\, dx = c x$$$ con $$$c=1$$$:

$$- \int{\frac{\cos{\left(2 x \right)}}{2} d x} + {\color{red}{\int{1 d x}}} = - \int{\frac{\cos{\left(2 x \right)}}{2} d x} + {\color{red}{x}}$$

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=\frac{1}{2}$$$ y $$$f{\left(x \right)} = \cos{\left(2 x \right)}$$$:

$$x - {\color{red}{\int{\frac{\cos{\left(2 x \right)}}{2} d x}}} = x - {\color{red}{\left(\frac{\int{\cos{\left(2 x \right)} d x}}{2}\right)}}$$

Sea $$$u=2 x$$$.

Entonces $$$du=\left(2 x\right)^{\prime }dx = 2 dx$$$ (los pasos pueden verse »), y obtenemos que $$$dx = \frac{du}{2}$$$.

La integral puede reescribirse como

$$x - \frac{{\color{red}{\int{\cos{\left(2 x \right)} d x}}}}{2} = x - \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}}{2}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{2}$$$ y $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$$x - \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}}{2} = x - \frac{{\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{2}\right)}}}{2}$$

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

$$x - \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{4} = x - \frac{{\color{red}{\sin{\left(u \right)}}}}{4}$$

Recordemos que $$$u=2 x$$$:

$$x - \frac{\sin{\left({\color{red}{u}} \right)}}{4} = x - \frac{\sin{\left({\color{red}{\left(2 x\right)}} \right)}}{4}$$

Por lo tanto,

$$\int{\left(1 - \frac{\cos{\left(2 x \right)}}{2}\right)d x} = x - \frac{\sin{\left(2 x \right)}}{4}$$

Añade la constante de integración:

$$\int{\left(1 - \frac{\cos{\left(2 x \right)}}{2}\right)d x} = x - \frac{\sin{\left(2 x \right)}}{4}+C$$

$$$\int \left(1 - \frac{\cos{\left(2 x \right)}}{2}\right)\, dx = \left(x - \frac{\sin{\left(2 x \right)}}{4}\right) + C$$$A