$$$20 x^{2} - \frac{2}{x}$$$の積分

$$$\int \left(20 x^{2} - \frac{2}{x}\right)\, dx$$$ を求めよ。

項別に積分せよ:

$${\color{red}{\int{\left(20 x^{2} - \frac{2}{x}\right)d x}}} = {\color{red}{\left(- \int{\frac{2}{x} d x} + \int{20 x^{2} d x}\right)}}$$

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=2$$$ と $$$f{\left(x \right)} = \frac{1}{x}$$$ に対して適用する:

$$\int{20 x^{2} d x} - {\color{red}{\int{\frac{2}{x} d x}}} = \int{20 x^{2} d x} - {\color{red}{\left(2 \int{\frac{1}{x} d x}\right)}}$$

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

$$\int{20 x^{2} d x} - 2 {\color{red}{\int{\frac{1}{x} d x}}} = \int{20 x^{2} d x} - 2 {\color{red}{\ln{\left(\left|{x}\right| \right)}}}$$

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=20$$$ と $$$f{\left(x \right)} = x^{2}$$$ に対して適用する:

$$- 2 \ln{\left(\left|{x}\right| \right)} + {\color{red}{\int{20 x^{2} d x}}} = - 2 \ln{\left(\left|{x}\right| \right)} + {\color{red}{\left(20 \int{x^{2} d x}\right)}}$$

$$$n=2$$$ を用いて、べき乗の法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ を適用します:

$$- 2 \ln{\left(\left|{x}\right| \right)} + 20 {\color{red}{\int{x^{2} d x}}}=- 2 \ln{\left(\left|{x}\right| \right)} + 20 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- 2 \ln{\left(\left|{x}\right| \right)} + 20 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

したがって、

$$\int{\left(20 x^{2} - \frac{2}{x}\right)d x} = \frac{20 x^{3}}{3} - 2 \ln{\left(\left|{x}\right| \right)}$$

積分定数を加える:

$$\int{\left(20 x^{2} - \frac{2}{x}\right)d x} = \frac{20 x^{3}}{3} - 2 \ln{\left(\left|{x}\right| \right)}+C$$

$$$\int \left(20 x^{2} - \frac{2}{x}\right)\, dx = \left(\frac{20 x^{3}}{3} - 2 \ln\left(\left|{x}\right|\right)\right) + C$$$A