Integral of $$$3^{- x}$$$

Find $$$\int 3^{- x}\, dx$$$.

Let $$$u=- x$$$.

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

Therefore,

$${\color{red}{\int{3^{- x} d x}}} = {\color{red}{\int{\left(- 3^{u}\right)d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=-1$$$ and $$$f{\left(u \right)} = 3^{u}$$$:

$${\color{red}{\int{\left(- 3^{u}\right)d u}}} = {\color{red}{\left(- \int{3^{u} d u}\right)}}$$

Apply the exponential rule $$$\int{a^{u} d u} = \frac{a^{u}}{\ln{\left(a \right)}}$$$ with $$$a=3$$$:

$$- {\color{red}{\int{3^{u} d u}}} = - {\color{red}{\frac{3^{u}}{\ln{\left(3 \right)}}}}$$

Recall that $$$u=- x$$$:

$$- \frac{3^{{\color{red}{u}}}}{\ln{\left(3 \right)}} = - \frac{3^{{\color{red}{\left(- x\right)}}}}{\ln{\left(3 \right)}}$$

Therefore,

$$\int{3^{- x} d x} = - \frac{3^{- x}}{\ln{\left(3 \right)}}$$

Add the constant of integration:

$$\int{3^{- x} d x} = - \frac{3^{- x}}{\ln{\left(3 \right)}}+C$$

$$$\int 3^{- x}\, dx = - \frac{3^{- x}}{\ln\left(3\right)} + C$$$A