Integral de $$$\frac{\sqrt{11} e^{- \frac{x}{2}}}{22}$$$

Halla $$$\int \frac{\sqrt{11} e^{- \frac{x}{2}}}{22}\, dx$$$.

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=\frac{\sqrt{11}}{22}$$$ y $$$f{\left(x \right)} = e^{- \frac{x}{2}}$$$:

$${\color{red}{\int{\frac{\sqrt{11} e^{- \frac{x}{2}}}{22} d x}}} = {\color{red}{\left(\frac{\sqrt{11} \int{e^{- \frac{x}{2}} d x}}{22}\right)}}$$

Sea $$$u=- \frac{x}{2}$$$.

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

Por lo tanto,

$$\frac{\sqrt{11} {\color{red}{\int{e^{- \frac{x}{2}} d x}}}}{22} = \frac{\sqrt{11} {\color{red}{\int{\left(- 2 e^{u}\right)d u}}}}{22}$$

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

$$\frac{\sqrt{11} {\color{red}{\int{\left(- 2 e^{u}\right)d u}}}}{22} = \frac{\sqrt{11} {\color{red}{\left(- 2 \int{e^{u} d u}\right)}}}{22}$$

La integral de la función exponencial es $$$\int{e^{u} d u} = e^{u}$$$:

$$- \frac{\sqrt{11} {\color{red}{\int{e^{u} d u}}}}{11} = - \frac{\sqrt{11} {\color{red}{e^{u}}}}{11}$$

Recordemos que $$$u=- \frac{x}{2}$$$:

$$- \frac{\sqrt{11} e^{{\color{red}{u}}}}{11} = - \frac{\sqrt{11} e^{{\color{red}{\left(- \frac{x}{2}\right)}}}}{11}$$

Por lo tanto,

$$\int{\frac{\sqrt{11} e^{- \frac{x}{2}}}{22} d x} = - \frac{\sqrt{11} e^{- \frac{x}{2}}}{11}$$

Añade la constante de integración:

$$\int{\frac{\sqrt{11} e^{- \frac{x}{2}}}{22} d x} = - \frac{\sqrt{11} e^{- \frac{x}{2}}}{11}+C$$

$$$\int \frac{\sqrt{11} e^{- \frac{x}{2}}}{22}\, dx = - \frac{\sqrt{11} e^{- \frac{x}{2}}}{11} + C$$$A