$$$\frac{2 \epsilon r^{2}}{5}$$$ 對 $$$\epsilon$$$ 的積分
您的輸入
求$$$\int \frac{2 \epsilon r^{2}}{5}\, d\epsilon$$$。
解答
套用常數倍法則 $$$\int c f{\left(\epsilon \right)}\, d\epsilon = c \int f{\left(\epsilon \right)}\, d\epsilon$$$，使用 $$$c=\frac{2 r^{2}}{5}$$$ 與 $$$f{\left(\epsilon \right)} = \epsilon$$$：
$${\color{red}{\int{\frac{2 \epsilon r^{2}}{5} d \epsilon}}} = {\color{red}{\left(\frac{2 r^{2} \int{\epsilon d \epsilon}}{5}\right)}}$$
套用冪次法則 $$$\int \epsilon^{n}\, d\epsilon = \frac{\epsilon^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=1$$$：
$$\frac{2 r^{2} {\color{red}{\int{\epsilon d \epsilon}}}}{5}=\frac{2 r^{2} {\color{red}{\frac{\epsilon^{1 + 1}}{1 + 1}}}}{5}=\frac{2 r^{2} {\color{red}{\left(\frac{\epsilon^{2}}{2}\right)}}}{5}$$
因此，
$$\int{\frac{2 \epsilon r^{2}}{5} d \epsilon} = \frac{\epsilon^{2} r^{2}}{5}$$
加上積分常數：
$$\int{\frac{2 \epsilon r^{2}}{5} d \epsilon} = \frac{\epsilon^{2} r^{2}}{5}+C$$
答案
$$$\int \frac{2 \epsilon r^{2}}{5}\, d\epsilon = \frac{\epsilon^{2} r^{2}}{5} + C$$$A