$$$\frac{a}{r^{10}} - \frac{b}{r^{5}}$$$ の $$$r$$$ に関する積分

$$$\int \left(\frac{a}{r^{10}} - \frac{b}{r^{5}}\right)\, dr$$$ を求めよ。

項別に積分せよ:

$${\color{red}{\int{\left(\frac{a}{r^{10}} - \frac{b}{r^{5}}\right)d r}}} = {\color{red}{\left(\int{\frac{a}{r^{10}} d r} - \int{\frac{b}{r^{5}} d r}\right)}}$$

定数倍の法則 $$$\int c f{\left(r \right)}\, dr = c \int f{\left(r \right)}\, dr$$$ を、$$$c=a$$$ と $$$f{\left(r \right)} = \frac{1}{r^{10}}$$$ に対して適用する:

$$- \int{\frac{b}{r^{5}} d r} + {\color{red}{\int{\frac{a}{r^{10}} d r}}} = - \int{\frac{b}{r^{5}} d r} + {\color{red}{a \int{\frac{1}{r^{10}} d r}}}$$

$$$n=-10$$$ を用いて、べき乗の法則 $$$\int r^{n}\, dr = \frac{r^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ を適用します:

$$a {\color{red}{\int{\frac{1}{r^{10}} d r}}} - \int{\frac{b}{r^{5}} d r}=a {\color{red}{\int{r^{-10} d r}}} - \int{\frac{b}{r^{5}} d r}=a {\color{red}{\frac{r^{-10 + 1}}{-10 + 1}}} - \int{\frac{b}{r^{5}} d r}=a {\color{red}{\left(- \frac{r^{-9}}{9}\right)}} - \int{\frac{b}{r^{5}} d r}=a {\color{red}{\left(- \frac{1}{9 r^{9}}\right)}} - \int{\frac{b}{r^{5}} d r}$$

定数倍の法則 $$$\int c f{\left(r \right)}\, dr = c \int f{\left(r \right)}\, dr$$$ を、$$$c=b$$$ と $$$f{\left(r \right)} = \frac{1}{r^{5}}$$$ に対して適用する:

$$- \frac{a}{9 r^{9}} - {\color{red}{\int{\frac{b}{r^{5}} d r}}} = - \frac{a}{9 r^{9}} - {\color{red}{b \int{\frac{1}{r^{5}} d r}}}$$

$$$n=-5$$$ を用いて、べき乗の法則 $$$\int r^{n}\, dr = \frac{r^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ を適用します:

$$- \frac{a}{9 r^{9}} - b {\color{red}{\int{\frac{1}{r^{5}} d r}}}=- \frac{a}{9 r^{9}} - b {\color{red}{\int{r^{-5} d r}}}=- \frac{a}{9 r^{9}} - b {\color{red}{\frac{r^{-5 + 1}}{-5 + 1}}}=- \frac{a}{9 r^{9}} - b {\color{red}{\left(- \frac{r^{-4}}{4}\right)}}=- \frac{a}{9 r^{9}} - b {\color{red}{\left(- \frac{1}{4 r^{4}}\right)}}$$

したがって、

$$\int{\left(\frac{a}{r^{10}} - \frac{b}{r^{5}}\right)d r} = - \frac{a}{9 r^{9}} + \frac{b}{4 r^{4}}$$

簡単化せよ:

$$\int{\left(\frac{a}{r^{10}} - \frac{b}{r^{5}}\right)d r} = \frac{- \frac{a}{9} + \frac{b r^{5}}{4}}{r^{9}}$$

積分定数を加える:

$$\int{\left(\frac{a}{r^{10}} - \frac{b}{r^{5}}\right)d r} = \frac{- \frac{a}{9} + \frac{b r^{5}}{4}}{r^{9}}+C$$

$$$\int \left(\frac{a}{r^{10}} - \frac{b}{r^{5}}\right)\, dr = \frac{- \frac{a}{9} + \frac{b r^{5}}{4}}{r^{9}} + C$$$A