Integral de $$$- 2 \csc^{3}{\left(x \right)} + \sec^{2}{\left(x \right)} - 1$$$

Halla $$$\int \left(- 2 \csc^{3}{\left(x \right)} + \sec^{2}{\left(x \right)} - 1\right)\, dx$$$.

Integra término a término:

$${\color{red}{\int{\left(- 2 \csc^{3}{\left(x \right)} + \sec^{2}{\left(x \right)} - 1\right)d x}}} = {\color{red}{\left(- \int{1 d x} - \int{2 \csc^{3}{\left(x \right)} d x} + \int{\sec^{2}{\left(x \right)} d x}\right)}}$$

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

$$- \int{2 \csc^{3}{\left(x \right)} d x} + \int{\sec^{2}{\left(x \right)} d x} - {\color{red}{\int{1 d x}}} = - \int{2 \csc^{3}{\left(x \right)} d x} + \int{\sec^{2}{\left(x \right)} d x} - {\color{red}{x}}$$

La integral de $$$\sec^{2}{\left(x \right)}$$$ es $$$\int{\sec^{2}{\left(x \right)} d x} = \tan{\left(x \right)}$$$:

$$- x - \int{2 \csc^{3}{\left(x \right)} d x} + {\color{red}{\int{\sec^{2}{\left(x \right)} d x}}} = - x - \int{2 \csc^{3}{\left(x \right)} d x} + {\color{red}{\tan{\left(x \right)}}}$$

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=2$$$ y $$$f{\left(x \right)} = \csc^{3}{\left(x \right)}$$$:

$$- x + \tan{\left(x \right)} - {\color{red}{\int{2 \csc^{3}{\left(x \right)} d x}}} = - x + \tan{\left(x \right)} - {\color{red}{\left(2 \int{\csc^{3}{\left(x \right)} d x}\right)}}$$

Para la integral $$$\int{\csc^{3}{\left(x \right)} d x}$$$, utiliza la integración por partes $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Sean $$$\operatorname{u}=\csc{\left(x \right)}$$$ y $$$\operatorname{dv}=\csc^{2}{\left(x \right)} dx$$$.

Entonces $$$\operatorname{du}=\left(\csc{\left(x \right)}\right)^{\prime }dx=- \cot{\left(x \right)} \csc{\left(x \right)} dx$$$ (los pasos pueden verse ») y $$$\operatorname{v}=\int{\csc^{2}{\left(x \right)} d x}=- \cot{\left(x \right)}$$$ (los pasos pueden verse »).

Entonces,

$$\int{\csc^{3}{\left(x \right)} d x}=\csc{\left(x \right)} \cdot \left(- \cot{\left(x \right)}\right)-\int{\left(- \cot{\left(x \right)}\right) \cdot \left(- \cot{\left(x \right)} \csc{\left(x \right)}\right) d x}=- \cot{\left(x \right)} \csc{\left(x \right)} - \int{\cot^{2}{\left(x \right)} \csc{\left(x \right)} d x}$$

Aplica la fórmula $$$\cot^{2}{\left(x \right)} = \csc^{2}{\left(x \right)} - 1$$$:

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

Expandir:

$$- \cot{\left(x \right)} \csc{\left(x \right)} - \int{\left(\csc^{2}{\left(x \right)} - 1\right) \csc{\left(x \right)} d x}=- \cot{\left(x \right)} \csc{\left(x \right)} - \int{\left(\csc^{3}{\left(x \right)} - \csc{\left(x \right)}\right)d x}$$

La integral de una diferencia es la diferencia de las integrales:

$$- \cot{\left(x \right)} \csc{\left(x \right)} - \int{\left(\csc^{3}{\left(x \right)} - \csc{\left(x \right)}\right)d x}=- \cot{\left(x \right)} \csc{\left(x \right)} + \int{\csc{\left(x \right)} d x} - \int{\csc^{3}{\left(x \right)} d x}$$

Por lo tanto, obtenemos la siguiente ecuación lineal simple con respecto a la integral:

$${\color{red}{\int{\csc^{3}{\left(x \right)} d x}}}=- \cot{\left(x \right)} \csc{\left(x \right)} + \int{\csc{\left(x \right)} d x} - {\color{red}{\int{\csc^{3}{\left(x \right)} d x}}}$$

Al resolverlo, obtenemos que

$$\int{\csc^{3}{\left(x \right)} d x}=- \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2} + \frac{\int{\csc{\left(x \right)} d x}}{2}$$

Por lo tanto,

$$- x + \tan{\left(x \right)} - 2 {\color{red}{\int{\csc^{3}{\left(x \right)} d x}}} = - x + \tan{\left(x \right)} - 2 {\color{red}{\left(- \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2} + \frac{\int{\csc{\left(x \right)} d x}}{2}\right)}}$$

Reescribe la cosecante como $$$\csc\left(x\right)=\frac{1}{\sin\left(x\right)}$$$:

$$- x + \tan{\left(x \right)} + \cot{\left(x \right)} \csc{\left(x \right)} - {\color{red}{\int{\csc{\left(x \right)} d x}}} = - x + \tan{\left(x \right)} + \cot{\left(x \right)} \csc{\left(x \right)} - {\color{red}{\int{\frac{1}{\sin{\left(x \right)}} d x}}}$$

Reescribe el seno utilizando la fórmula del ángulo doble $$$\sin\left(x\right)=2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)$$$:

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

Multiplica el numerador y el denominador por $$$\sec^2\left(\frac{x}{2} \right)$$$:

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

Sea $$$u=\tan{\left(\frac{x}{2} \right)}$$$.

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

Por lo tanto,

$$- x + \tan{\left(x \right)} + \cot{\left(x \right)} \csc{\left(x \right)} - {\color{red}{\int{\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{2 \tan{\left(\frac{x}{2} \right)}} d x}}} = - x + \tan{\left(x \right)} + \cot{\left(x \right)} \csc{\left(x \right)} - {\color{red}{\int{\frac{1}{u} d u}}}$$

La integral de $$$\frac{1}{u}$$$ es $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

$$- x + \tan{\left(x \right)} + \cot{\left(x \right)} \csc{\left(x \right)} - {\color{red}{\int{\frac{1}{u} d u}}} = - x + \tan{\left(x \right)} + \cot{\left(x \right)} \csc{\left(x \right)} - {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

Recordemos que $$$u=\tan{\left(\frac{x}{2} \right)}$$$:

$$- x - \ln{\left(\left|{{\color{red}{u}}}\right| \right)} + \tan{\left(x \right)} + \cot{\left(x \right)} \csc{\left(x \right)} = - x - \ln{\left(\left|{{\color{red}{\tan{\left(\frac{x}{2} \right)}}}}\right| \right)} + \tan{\left(x \right)} + \cot{\left(x \right)} \csc{\left(x \right)}$$

Por lo tanto,

$$\int{\left(- 2 \csc^{3}{\left(x \right)} + \sec^{2}{\left(x \right)} - 1\right)d x} = - x - \ln{\left(\left|{\tan{\left(\frac{x}{2} \right)}}\right| \right)} + \tan{\left(x \right)} + \cot{\left(x \right)} \csc{\left(x \right)}$$

Añade la constante de integración:

$$\int{\left(- 2 \csc^{3}{\left(x \right)} + \sec^{2}{\left(x \right)} - 1\right)d x} = - x - \ln{\left(\left|{\tan{\left(\frac{x}{2} \right)}}\right| \right)} + \tan{\left(x \right)} + \cot{\left(x \right)} \csc{\left(x \right)}+C$$

$$$\int \left(- 2 \csc^{3}{\left(x \right)} + \sec^{2}{\left(x \right)} - 1\right)\, dx = \left(- x - \ln\left(\left|{\tan{\left(\frac{x}{2} \right)}}\right|\right) + \tan{\left(x \right)} + \cot{\left(x \right)} \csc{\left(x \right)}\right) + C$$$A