Integral de $$$45 e^{- \frac{t}{20}}$$$

Encontre $$$\int 45 e^{- \frac{t}{20}}\, dt$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ usando $$$c=45$$$ e $$$f{\left(t \right)} = e^{- \frac{t}{20}}$$$:

$${\color{red}{\int{45 e^{- \frac{t}{20}} d t}}} = {\color{red}{\left(45 \int{e^{- \frac{t}{20}} d t}\right)}}$$

Seja $$$u=- \frac{t}{20}$$$.

Então $$$du=\left(- \frac{t}{20}\right)^{\prime }dt = - \frac{dt}{20}$$$ (veja os passos »), e obtemos $$$dt = - 20 du$$$.

Logo,

$$45 {\color{red}{\int{e^{- \frac{t}{20}} d t}}} = 45 {\color{red}{\int{\left(- 20 e^{u}\right)d u}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=-20$$$ e $$$f{\left(u \right)} = e^{u}$$$:

$$45 {\color{red}{\int{\left(- 20 e^{u}\right)d u}}} = 45 {\color{red}{\left(- 20 \int{e^{u} d u}\right)}}$$

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

$$- 900 {\color{red}{\int{e^{u} d u}}} = - 900 {\color{red}{e^{u}}}$$

Recorde que $$$u=- \frac{t}{20}$$$:

$$- 900 e^{{\color{red}{u}}} = - 900 e^{{\color{red}{\left(- \frac{t}{20}\right)}}}$$

Portanto,

$$\int{45 e^{- \frac{t}{20}} d t} = - 900 e^{- \frac{t}{20}}$$

Adicione a constante de integração:

$$\int{45 e^{- \frac{t}{20}} d t} = - 900 e^{- \frac{t}{20}}+C$$

$$$\int 45 e^{- \frac{t}{20}}\, dt = - 900 e^{- \frac{t}{20}} + C$$$A