Integral de $$$e^{\frac{t}{50}}$$$

Halla $$$\int e^{\frac{t}{50}}\, dt$$$.

Sea $$$u=\frac{t}{50}$$$.

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

La integral puede reescribirse como

$${\color{red}{\int{e^{\frac{t}{50}} d t}}} = {\color{red}{\int{50 e^{u} d u}}}$$

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

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

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

$$50 {\color{red}{\int{e^{u} d u}}} = 50 {\color{red}{e^{u}}}$$

Recordemos que $$$u=\frac{t}{50}$$$:

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

Por lo tanto,

$$\int{e^{\frac{t}{50}} d t} = 50 e^{\frac{t}{50}}$$

Añade la constante de integración:

$$\int{e^{\frac{t}{50}} d t} = 50 e^{\frac{t}{50}}+C$$

$$$\int e^{\frac{t}{50}}\, dt = 50 e^{\frac{t}{50}} + C$$$A