Integral de $$$\frac{4}{t}$$$

Encontre $$$\int \frac{4}{t}\, dt$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ usando $$$c=4$$$ e $$$f{\left(t \right)} = \frac{1}{t}$$$:

$${\color{red}{\int{\frac{4}{t} d t}}} = {\color{red}{\left(4 \int{\frac{1}{t} d t}\right)}}$$

A integral de $$$\frac{1}{t}$$$ é $$$\int{\frac{1}{t} d t} = \ln{\left(\left|{t}\right| \right)}$$$:

$$4 {\color{red}{\int{\frac{1}{t} d t}}} = 4 {\color{red}{\ln{\left(\left|{t}\right| \right)}}}$$

Portanto,

$$\int{\frac{4}{t} d t} = 4 \ln{\left(\left|{t}\right| \right)}$$

Adicione a constante de integração:

$$\int{\frac{4}{t} d t} = 4 \ln{\left(\left|{t}\right| \right)}+C$$

$$$\int \frac{4}{t}\, dt = 4 \ln\left(\left|{t}\right|\right) + C$$$A