Integral de $$$4 e^{2 x}$$$

Halla $$$\int 4 e^{2 x}\, dx$$$.

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=4$$$ y $$$f{\left(x \right)} = e^{2 x}$$$:

$${\color{red}{\int{4 e^{2 x} d x}}} = {\color{red}{\left(4 \int{e^{2 x} d x}\right)}}$$

Sea $$$u=2 x$$$.

Entonces $$$du=\left(2 x\right)^{\prime }dx = 2 dx$$$ (los pasos pueden verse »), y obtenemos que $$$dx = \frac{du}{2}$$$.

Por lo tanto,

$$4 {\color{red}{\int{e^{2 x} d x}}} = 4 {\color{red}{\int{\frac{e^{u}}{2} d u}}}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{2}$$$ y $$$f{\left(u \right)} = e^{u}$$$:

$$4 {\color{red}{\int{\frac{e^{u}}{2} d u}}} = 4 {\color{red}{\left(\frac{\int{e^{u} d u}}{2}\right)}}$$

La integral de la función exponencial es $$$\int{e^{u} d u} = e^{u}$$$:

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

Recordemos que $$$u=2 x$$$:

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

Por lo tanto,

$$\int{4 e^{2 x} d x} = 2 e^{2 x}$$

Añade la constante de integración:

$$\int{4 e^{2 x} d x} = 2 e^{2 x}+C$$

$$$\int 4 e^{2 x}\, dx = 2 e^{2 x} + C$$$A