Integral de $$$2 \sin^{2}{\left(x \right)} - 1$$$

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

Integra término a término:

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

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

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

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

$$- x + {\color{red}{\int{2 \sin^{2}{\left(x \right)} d x}}} = - x + {\color{red}{\left(2 \int{\sin^{2}{\left(x \right)} d x}\right)}}$$

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

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

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 $$$f{\left(x \right)} = 1 - \cos{\left(2 x \right)}$$$:

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

Integra término a término:

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

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

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

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

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

Por lo tanto,

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

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

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

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

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

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

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

Por lo tanto,

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

Añade la constante de integración:

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

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