$$$\frac{s^{2}}{d t}$$$ の $$$t$$$ に関する積分
入力内容
$$$\int \frac{s^{2}}{d t}\, dt$$$ を求めよ。
解答
定数倍の法則 $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ を、$$$c=\frac{s^{2}}{d}$$$ と $$$f{\left(t \right)} = \frac{1}{t}$$$ に対して適用する:
$${\color{red}{\int{\frac{s^{2}}{d t} d t}}} = {\color{red}{\frac{s^{2} \int{\frac{1}{t} d t}}{d}}}$$
$$$\frac{1}{t}$$$ の不定積分は $$$\int{\frac{1}{t} d t} = \ln{\left(\left|{t}\right| \right)}$$$ です:
$$\frac{s^{2} {\color{red}{\int{\frac{1}{t} d t}}}}{d} = \frac{s^{2} {\color{red}{\ln{\left(\left|{t}\right| \right)}}}}{d}$$
したがって、
$$\int{\frac{s^{2}}{d t} d t} = \frac{s^{2} \ln{\left(\left|{t}\right| \right)}}{d}$$
積分定数を加える:
$$\int{\frac{s^{2}}{d t} d t} = \frac{s^{2} \ln{\left(\left|{t}\right| \right)}}{d}+C$$
解答
$$$\int \frac{s^{2}}{d t}\, dt = \frac{s^{2} \ln\left(\left|{t}\right|\right)}{d} + C$$$A