Integral de $$$5^{- x}$$$

Halla $$$\int 5^{- x}\, dx$$$.

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

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

La integral se convierte en

$${\color{red}{\int{5^{- x} d x}}} = {\color{red}{\int{\left(- 5^{u}\right)d u}}}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=-1$$$ y $$$f{\left(u \right)} = 5^{u}$$$:

$${\color{red}{\int{\left(- 5^{u}\right)d u}}} = {\color{red}{\left(- \int{5^{u} d u}\right)}}$$

Apply the exponential rule $$$\int{a^{u} d u} = \frac{a^{u}}{\ln{\left(a \right)}}$$$ with $$$a=5$$$:

$$- {\color{red}{\int{5^{u} d u}}} = - {\color{red}{\frac{5^{u}}{\ln{\left(5 \right)}}}}$$

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

$$- \frac{5^{{\color{red}{u}}}}{\ln{\left(5 \right)}} = - \frac{5^{{\color{red}{\left(- x\right)}}}}{\ln{\left(5 \right)}}$$

Por lo tanto,

$$\int{5^{- x} d x} = - \frac{5^{- x}}{\ln{\left(5 \right)}}$$

Añade la constante de integración:

$$\int{5^{- x} d x} = - \frac{5^{- x}}{\ln{\left(5 \right)}}+C$$

$$$\int 5^{- x}\, dx = - \frac{5^{- x}}{\ln\left(5\right)} + C$$$A