$$$\frac{1}{126 t}$$$の積分

$$$\int \frac{1}{126 t}\, dt$$$ 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、$$$c=\frac{1}{126}$$$ と $$$f{\left(t \right)} = \frac{1}{t}$$$ に対して適用する:

$${\color{red}{\int{\frac{1}{126 t} d t}}} = {\color{red}{\left(\frac{\int{\frac{1}{t} d t}}{126}\right)}}$$

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

$$\frac{{\color{red}{\int{\frac{1}{t} d t}}}}{126} = \frac{{\color{red}{\ln{\left(\left|{t}\right| \right)}}}}{126}$$

したがって、

$$\int{\frac{1}{126 t} d t} = \frac{\ln{\left(\left|{t}\right| \right)}}{126}$$

積分定数を加える:

$$\int{\frac{1}{126 t} d t} = \frac{\ln{\left(\left|{t}\right| \right)}}{126}+C$$

$$$\int \frac{1}{126 t}\, dt = \frac{\ln\left(\left|{t}\right|\right)}{126} + C$$$A