Integral of $$$6^{- 3 x}$$$

Find $$$\int 6^{- 3 x}\, dx$$$.

The input is rewritten: $$$\int{6^{- 3 x} d x}=\int{216^{- x} d x}$$$.

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

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

Therefore,

$${\color{red}{\int{216^{- x} d x}}} = {\color{red}{\int{\left(- 216^{u}\right)d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=-1$$$ and $$$f{\left(u \right)} = 216^{u}$$$:

$${\color{red}{\int{\left(- 216^{u}\right)d u}}} = {\color{red}{\left(- \int{216^{u} d u}\right)}}$$

Apply the exponential rule $$$\int{a^{u} d u} = \frac{a^{u}}{\ln{\left(a \right)}}$$$ with $$$a=216$$$:

$$- {\color{red}{\int{216^{u} d u}}} = - {\color{red}{\frac{216^{u}}{\ln{\left(216 \right)}}}}$$

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

$$- \frac{216^{{\color{red}{u}}}}{\ln{\left(216 \right)}} = - \frac{216^{{\color{red}{\left(- x\right)}}}}{\ln{\left(216 \right)}}$$

Therefore,

$$\int{216^{- x} d x} = - \frac{216^{- x}}{\ln{\left(216 \right)}}$$

Simplify:

$$\int{216^{- x} d x} = - \frac{216^{- x}}{3 \ln{\left(6 \right)}}$$

Add the constant of integration:

$$\int{216^{- x} d x} = - \frac{216^{- x}}{3 \ln{\left(6 \right)}}+C$$

$$$\int 6^{- 3 x}\, dx = - \frac{216^{- x}}{3 \ln\left(6\right)} + C$$$A