Integral de $$$2 e^{3 x}$$$

Halla $$$\int 2 e^{3 x}\, dx$$$.

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=2$$$ y $$$f{\left(x \right)} = e^{3 x}$$$:

$${\color{red}{\int{2 e^{3 x} d x}}} = {\color{red}{\left(2 \int{e^{3 x} d x}\right)}}$$

Sea $$$u=3 x$$$.

Entonces $$$du=\left(3 x\right)^{\prime }dx = 3 dx$$$ (los pasos pueden verse »), y obtenemos que $$$dx = \frac{du}{3}$$$.

Entonces,

$$2 {\color{red}{\int{e^{3 x} d x}}} = 2 {\color{red}{\int{\frac{e^{u}}{3} d u}}}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{3}$$$ y $$$f{\left(u \right)} = e^{u}$$$:

$$2 {\color{red}{\int{\frac{e^{u}}{3} d u}}} = 2 {\color{red}{\left(\frac{\int{e^{u} d u}}{3}\right)}}$$

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

$$\frac{2 {\color{red}{\int{e^{u} d u}}}}{3} = \frac{2 {\color{red}{e^{u}}}}{3}$$

Recordemos que $$$u=3 x$$$:

$$\frac{2 e^{{\color{red}{u}}}}{3} = \frac{2 e^{{\color{red}{\left(3 x\right)}}}}{3}$$

Por lo tanto,

$$\int{2 e^{3 x} d x} = \frac{2 e^{3 x}}{3}$$

Añade la constante de integración:

$$\int{2 e^{3 x} d x} = \frac{2 e^{3 x}}{3}+C$$

$$$\int 2 e^{3 x}\, dx = \frac{2 e^{3 x}}{3} + C$$$A