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

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

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

Let $$$u=2 x$$$.

Then $$$du=\left(2 x\right)^{\prime }dx = 2 dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{2}$$$.

The integral becomes

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

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{2}$$$ and $$$f{\left(u \right)} = e^{u}$$$:

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

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 $$$u=2 x$$$:

$$e^{{\color{red}{u}}} = e^{{\color{red}{\left(2 x\right)}}}$$

Therefore,

$$\int{2 e^{2 x} d x} = e^{2 x}$$

Add the constant of integration:

$$\int{2 e^{2 x} d x} = e^{2 x}+C$$

Answer: $$$\int{2 e^{2 x} d x}=e^{2 x}+C$$$