Integral of $$$e^{- a}$$$

Find $$$\int e^{- a}\, da$$$.

Let $$$u=- a$$$.

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

The integral can be rewritten as

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

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

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

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

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

Recall that $$$u=- a$$$:

$$- e^{{\color{red}{u}}} = - e^{{\color{red}{\left(- a\right)}}}$$

Therefore,

$$\int{e^{- a} d a} = - e^{- a}$$

Add the constant of integration:

$$\int{e^{- a} d a} = - e^{- a}+C$$

$$$\int e^{- a}\, da = - e^{- a} + C$$$A