Integral de $$$\frac{2 x^{2}}{1 - x}$$$

Halla $$$\int \frac{2 x^{2}}{1 - x}\, dx$$$.

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

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

Como el grado del numerador no es menor que el grado del denominador, realiza la división larga de polinomios (los pasos pueden verse »):

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

Integra término a término:

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

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

$$- 2 \int{x d x} + 2 \int{\frac{1}{1 - x} d x} - 2 {\color{red}{\int{1 d x}}} = - 2 \int{x d x} + 2 \int{\frac{1}{1 - x} d x} - 2 {\color{red}{x}}$$

Sea $$$u=1 - x$$$.

Entonces $$$du=\left(1 - x\right)^{\prime }dx = - dx$$$ (los pasos pueden verse »), y obtenemos que $$$dx = - du$$$.

La integral se convierte en

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

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

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

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

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

Recordemos que $$$u=1 - x$$$:

$$- 2 x - 2 \ln{\left(\left|{{\color{red}{u}}}\right| \right)} - 2 \int{x d x} = - 2 x - 2 \ln{\left(\left|{{\color{red}{\left(1 - x\right)}}}\right| \right)} - 2 \int{x d x}$$

Aplica la regla de la potencia $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=1$$$:

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

Por lo tanto,

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

Añade la constante de integración:

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

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