Integral de $$$15 \sqrt{5} e^{5 x}$$$

Halla $$$\int 15 \sqrt{5} e^{5 x}\, dx$$$.

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=15 \sqrt{5}$$$ y $$$f{\left(x \right)} = e^{5 x}$$$:

$${\color{red}{\int{15 \sqrt{5} e^{5 x} d x}}} = {\color{red}{\left(15 \sqrt{5} \int{e^{5 x} d x}\right)}}$$

Sea $$$u=5 x$$$.

Entonces $$$du=\left(5 x\right)^{\prime }dx = 5 dx$$$ (los pasos pueden verse »), y obtenemos que $$$dx = \frac{du}{5}$$$.

Por lo tanto,

$$15 \sqrt{5} {\color{red}{\int{e^{5 x} d x}}} = 15 \sqrt{5} {\color{red}{\int{\frac{e^{u}}{5} d u}}}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{5}$$$ y $$$f{\left(u \right)} = e^{u}$$$:

$$15 \sqrt{5} {\color{red}{\int{\frac{e^{u}}{5} d u}}} = 15 \sqrt{5} {\color{red}{\left(\frac{\int{e^{u} d u}}{5}\right)}}$$

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

$$3 \sqrt{5} {\color{red}{\int{e^{u} d u}}} = 3 \sqrt{5} {\color{red}{e^{u}}}$$

Recordemos que $$$u=5 x$$$:

$$3 \sqrt{5} e^{{\color{red}{u}}} = 3 \sqrt{5} e^{{\color{red}{\left(5 x\right)}}}$$

Por lo tanto,

$$\int{15 \sqrt{5} e^{5 x} d x} = 3 \sqrt{5} e^{5 x}$$

Añade la constante de integración:

$$\int{15 \sqrt{5} e^{5 x} d x} = 3 \sqrt{5} e^{5 x}+C$$

$$$\int 15 \sqrt{5} e^{5 x}\, dx = 3 \sqrt{5} e^{5 x} + C$$$A