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

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

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

Então $$$du=\left(\frac{t}{50}\right)^{\prime }dt = \frac{dt}{50}$$$ (veja os passos »), e obtemos $$$dt = 50 du$$$.

Logo,

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

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=50$$$ e $$$f{\left(u \right)} = e^{u}$$$:

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

A integral da função exponencial é $$$\int{e^{u} d u} = e^{u}$$$:

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

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

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

Portanto,

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

Adicione a constante de integração:

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

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