Integral de $$$\cos^{2}{\left(y \right)}$$$

Halla $$$\int \cos^{2}{\left(y \right)}\, dy$$$.

Aplica la fórmula de reducción de potencia $$$\cos^{2}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{1}{2}$$$ con $$$\alpha=y$$$:

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

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

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

Integra término a término:

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

Aplica la regla de la constante $$$\int c\, dy = c y$$$ con $$$c=1$$$:

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

Sea $$$u=2 y$$$.

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

La integral puede reescribirse como

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

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 $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

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

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

$$\frac{y}{2} + \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{4} = \frac{y}{2} + \frac{{\color{red}{\sin{\left(u \right)}}}}{4}$$

Recordemos que $$$u=2 y$$$:

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

Por lo tanto,

$$\int{\cos^{2}{\left(y \right)} d y} = \frac{y}{2} + \frac{\sin{\left(2 y \right)}}{4}$$

Añade la constante de integración:

$$\int{\cos^{2}{\left(y \right)} d y} = \frac{y}{2} + \frac{\sin{\left(2 y \right)}}{4}+C$$

$$$\int \cos^{2}{\left(y \right)}\, dy = \left(\frac{y}{2} + \frac{\sin{\left(2 y \right)}}{4}\right) + C$$$A