Integral de $$$e^{\frac{x}{3}}$$$

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

Sea $$$u=\frac{x}{3}$$$.

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

La integral se convierte en

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

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

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

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

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

Recordemos que $$$u=\frac{x}{3}$$$:

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

Por lo tanto,

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

Añade la constante de integración:

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

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