Integral of $$$e^{a x}$$$ with respect to $$$x$$$

Let $$$u=a x$$$.

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

The integral becomes

$${\color{red}{\int{e^{a x} d x}}} = {\color{red}{\int{\frac{e^{u}}{a} d u}}}$$

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

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

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

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

Recall that $$$u=a x$$$:

$$\frac{e^{{\color{red}{u}}}}{a} = \frac{e^{{\color{red}{a x}}}}{a}$$

Therefore,

$$\int{e^{a x} d x} = \frac{e^{a x}}{a}$$

Add the constant of integration:

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

$$$\int e^{a x}\, dx = \frac{e^{a x}}{a} + C$$$A