Integral de $$$70 e^{\frac{3 x}{50}}$$$

Halla $$$\int 70 e^{\frac{3 x}{50}}\, dx$$$.

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=70$$$ y $$$f{\left(x \right)} = e^{\frac{3 x}{50}}$$$:

$${\color{red}{\int{70 e^{\frac{3 x}{50}} d x}}} = {\color{red}{\left(70 \int{e^{\frac{3 x}{50}} d x}\right)}}$$

Sea $$$u=\frac{3 x}{50}$$$.

Entonces $$$du=\left(\frac{3 x}{50}\right)^{\prime }dx = \frac{3 dx}{50}$$$ (los pasos pueden verse »), y obtenemos que $$$dx = \frac{50 du}{3}$$$.

Entonces,

$$70 {\color{red}{\int{e^{\frac{3 x}{50}} d x}}} = 70 {\color{red}{\int{\frac{50 e^{u}}{3} d u}}}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{50}{3}$$$ y $$$f{\left(u \right)} = e^{u}$$$:

$$70 {\color{red}{\int{\frac{50 e^{u}}{3} d u}}} = 70 {\color{red}{\left(\frac{50 \int{e^{u} d u}}{3}\right)}}$$

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

$$\frac{3500 {\color{red}{\int{e^{u} d u}}}}{3} = \frac{3500 {\color{red}{e^{u}}}}{3}$$

Recordemos que $$$u=\frac{3 x}{50}$$$:

$$\frac{3500 e^{{\color{red}{u}}}}{3} = \frac{3500 e^{{\color{red}{\left(\frac{3 x}{50}\right)}}}}{3}$$

Por lo tanto,

$$\int{70 e^{\frac{3 x}{50}} d x} = \frac{3500 e^{\frac{3 x}{50}}}{3}$$

Añade la constante de integración:

$$\int{70 e^{\frac{3 x}{50}} d x} = \frac{3500 e^{\frac{3 x}{50}}}{3}+C$$

$$$\int 70 e^{\frac{3 x}{50}}\, dx = \frac{3500 e^{\frac{3 x}{50}}}{3} + C$$$A