Integral of $$$5^{- n}$$$

Find $$$\int 5^{- n}\, dn$$$.

Let $$$u=- n$$$.

Then $$$du=\left(- n\right)^{\prime }dn = - dn$$$ (steps can be seen »), and we have that $$$dn = - du$$$.

The integral becomes

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

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=-1$$$ and $$$f{\left(u \right)} = 5^{u}$$$:

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

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

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

Recall that $$$u=- n$$$:

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

Therefore,

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

Add the constant of integration:

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

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