Integral de $$$e^{a x}$$$ con respecto a $$$x$$$

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

Sea $$$u=a x$$$.

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

La integral se convierte en

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

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

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

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

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

Recordemos que $$$u=a x$$$:

$$\frac{e^{{\color{red}{u}}}}{a} = \frac{e^{{\color{red}{a x}}}}{a}$$

Por lo tanto,

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

Añade la constante de integración:

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

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