Integral de $$$36 \cos^{2}{\left(\theta \right)}$$$

Halla $$$\int 36 \cos^{2}{\left(\theta \right)}\, d\theta$$$.

Aplica la regla del factor constante $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$ con $$$c=36$$$ y $$$f{\left(\theta \right)} = \cos^{2}{\left(\theta \right)}$$$:

$${\color{red}{\int{36 \cos^{2}{\left(\theta \right)} d \theta}}} = {\color{red}{\left(36 \int{\cos^{2}{\left(\theta \right)} d \theta}\right)}}$$

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

$$36 {\color{red}{\int{\cos^{2}{\left(\theta \right)} d \theta}}} = 36 {\color{red}{\int{\left(\frac{\cos{\left(2 \theta \right)}}{2} + \frac{1}{2}\right)d \theta}}}$$

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

$$36 {\color{red}{\int{\left(\frac{\cos{\left(2 \theta \right)}}{2} + \frac{1}{2}\right)d \theta}}} = 36 {\color{red}{\left(\frac{\int{\left(\cos{\left(2 \theta \right)} + 1\right)d \theta}}{2}\right)}}$$

Integra término a término:

$$18 {\color{red}{\int{\left(\cos{\left(2 \theta \right)} + 1\right)d \theta}}} = 18 {\color{red}{\left(\int{1 d \theta} + \int{\cos{\left(2 \theta \right)} d \theta}\right)}}$$

Aplica la regla de la constante $$$\int c\, d\theta = c \theta$$$ con $$$c=1$$$:

$$18 \int{\cos{\left(2 \theta \right)} d \theta} + 18 {\color{red}{\int{1 d \theta}}} = 18 \int{\cos{\left(2 \theta \right)} d \theta} + 18 {\color{red}{\theta}}$$

Sea $$$u=2 \theta$$$.

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

La integral se convierte en

$$18 \theta + 18 {\color{red}{\int{\cos{\left(2 \theta \right)} d \theta}}} = 18 \theta + 18 {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}$$

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

$$18 \theta + 18 {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}} = 18 \theta + 18 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{2}\right)}}$$

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

$$18 \theta + 9 {\color{red}{\int{\cos{\left(u \right)} d u}}} = 18 \theta + 9 {\color{red}{\sin{\left(u \right)}}}$$

Recordemos que $$$u=2 \theta$$$:

$$18 \theta + 9 \sin{\left({\color{red}{u}} \right)} = 18 \theta + 9 \sin{\left({\color{red}{\left(2 \theta\right)}} \right)}$$

Por lo tanto,

$$\int{36 \cos^{2}{\left(\theta \right)} d \theta} = 18 \theta + 9 \sin{\left(2 \theta \right)}$$

Añade la constante de integración:

$$\int{36 \cos^{2}{\left(\theta \right)} d \theta} = 18 \theta + 9 \sin{\left(2 \theta \right)}+C$$

$$$\int 36 \cos^{2}{\left(\theta \right)}\, d\theta = \left(18 \theta + 9 \sin{\left(2 \theta \right)}\right) + C$$$A