Integral de $$$e^{8 x}$$$

Halla $$$\int e^{8 x}\, dx$$$.

Sea $$$u=8 x$$$.

Entonces $$$du=\left(8 x\right)^{\prime }dx = 8 dx$$$ (los pasos pueden verse »), y obtenemos que $$$dx = \frac{du}{8}$$$.

La integral se convierte en

$${\color{red}{\int{e^{8 x} d x}}} = {\color{red}{\int{\frac{e^{u}}{8} d u}}}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{8}$$$ y $$$f{\left(u \right)} = e^{u}$$$:

$${\color{red}{\int{\frac{e^{u}}{8} d u}}} = {\color{red}{\left(\frac{\int{e^{u} d u}}{8}\right)}}$$

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

$$\frac{{\color{red}{\int{e^{u} d u}}}}{8} = \frac{{\color{red}{e^{u}}}}{8}$$

Recordemos que $$$u=8 x$$$:

$$\frac{e^{{\color{red}{u}}}}{8} = \frac{e^{{\color{red}{\left(8 x\right)}}}}{8}$$

Por lo tanto,

$$\int{e^{8 x} d x} = \frac{e^{8 x}}{8}$$

Añade la constante de integración:

$$\int{e^{8 x} d x} = \frac{e^{8 x}}{8}+C$$

$$$\int e^{8 x}\, dx = \frac{e^{8 x}}{8} + C$$$A