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

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

Simplificar el integrando:

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

Integra término a término:

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

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

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

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

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

Por lo tanto,

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

Simplificar:

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

Añade la constante de integración:

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

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