$$$\frac{3}{t}$$$の積分

$$$\int \frac{3}{t}\, dt$$$ を求めよ。

定数倍の法則 $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ を、$$$c=3$$$ と $$$f{\left(t \right)} = \frac{1}{t}$$$ に対して適用する:

$${\color{red}{\int{\frac{3}{t} d t}}} = {\color{red}{\left(3 \int{\frac{1}{t} d t}\right)}}$$

$$$\frac{1}{t}$$$ の不定積分は $$$\int{\frac{1}{t} d t} = \ln{\left(\left|{t}\right| \right)}$$$ です:

$$3 {\color{red}{\int{\frac{1}{t} d t}}} = 3 {\color{red}{\ln{\left(\left|{t}\right| \right)}}}$$

したがって、

$$\int{\frac{3}{t} d t} = 3 \ln{\left(\left|{t}\right| \right)}$$

積分定数を加える:

$$\int{\frac{3}{t} d t} = 3 \ln{\left(\left|{t}\right| \right)}+C$$

$$$\int \frac{3}{t}\, dt = 3 \ln\left(\left|{t}\right|\right) + C$$$A