Integral of $$$1 - 2 e^{x}$$$

Find $$$\int \left(1 - 2 e^{x}\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(1 - 2 e^{x}\right)d x}}} = {\color{red}{\left(\int{1 d x} - \int{2 e^{x} d x}\right)}}$$

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=1$$$:

$$- \int{2 e^{x} d x} + {\color{red}{\int{1 d x}}} = - \int{2 e^{x} d x} + {\color{red}{x}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=2$$$ and $$$f{\left(x \right)} = e^{x}$$$:

$$x - {\color{red}{\int{2 e^{x} d x}}} = x - {\color{red}{\left(2 \int{e^{x} d x}\right)}}$$

The integral of the exponential function is $$$\int{e^{x} d x} = e^{x}$$$:

$$x - 2 {\color{red}{\int{e^{x} d x}}} = x - 2 {\color{red}{e^{x}}}$$

Therefore,

$$\int{\left(1 - 2 e^{x}\right)d x} = x - 2 e^{x}$$

Add the constant of integration:

$$\int{\left(1 - 2 e^{x}\right)d x} = x - 2 e^{x}+C$$

$$$\int \left(1 - 2 e^{x}\right)\, dx = \left(x - 2 e^{x}\right) + C$$$A