Integral of $$$- k^{x}$$$ with respect to $$$x$$$

Find $$$\int \left(- k^{x}\right)\, dx$$$.

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

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

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

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

Therefore,

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

Add the constant of integration:

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

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