$$$\frac{s^{2}}{d t}$$$ 對 $$$t$$$ 的積分

求$$$\int \frac{s^{2}}{d t}\, dt$$$。

套用常數倍法則 $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$，使用 $$$c=\frac{s^{2}}{d}$$$ 與 $$$f{\left(t \right)} = \frac{1}{t}$$$：

$${\color{red}{\int{\frac{s^{2}}{d t} d t}}} = {\color{red}{\frac{s^{2} \int{\frac{1}{t} d t}}{d}}}$$

$$$\frac{1}{t}$$$ 的積分是 $$$\int{\frac{1}{t} d t} = \ln{\left(\left|{t}\right| \right)}$$$：

$$\frac{s^{2} {\color{red}{\int{\frac{1}{t} d t}}}}{d} = \frac{s^{2} {\color{red}{\ln{\left(\left|{t}\right| \right)}}}}{d}$$

因此，

$$\int{\frac{s^{2}}{d t} d t} = \frac{s^{2} \ln{\left(\left|{t}\right| \right)}}{d}$$

加上積分常數：

$$\int{\frac{s^{2}}{d t} d t} = \frac{s^{2} \ln{\left(\left|{t}\right| \right)}}{d}+C$$

$$$\int \frac{s^{2}}{d t}\, dt = \frac{s^{2} \ln\left(\left|{t}\right|\right)}{d} + C$$$A