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

Find $$$\int 2^{x} e^{x}\, dx$$$.

The input is rewritten: $$$\int{2^{x} e^{x} d x}=\int{\left(2 e\right)^{x} d x}$$$.

Apply the exponential rule $$$\int{a^{x} d x} = \frac{a^{x}}{\ln{\left(a \right)}}$$$ with $$$a=2 e$$$:

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

Therefore,

$$\int{\left(2 e\right)^{x} d x} = \frac{\left(2 e\right)^{x}}{\ln{\left(2 e \right)}}$$

Simplify:

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

Add the constant of integration:

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

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