$$$- 2 x^{2} + \frac{x}{57}$$$の積分

$$$\int \left(- 2 x^{2} + \frac{x}{57}\right)\, dx$$$ を求めよ。

項別に積分せよ:

$${\color{red}{\int{\left(- 2 x^{2} + \frac{x}{57}\right)d x}}} = {\color{red}{\left(\int{\frac{x}{57} d x} - \int{2 x^{2} d x}\right)}}$$

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

$$\int{\frac{x}{57} d x} - {\color{red}{\int{2 x^{2} d x}}} = \int{\frac{x}{57} d x} - {\color{red}{\left(2 \int{x^{2} d x}\right)}}$$

$$$n=2$$$ 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:

$$\int{\frac{x}{57} d x} - 2 {\color{red}{\int{x^{2} d x}}}=\int{\frac{x}{57} d x} - 2 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=\int{\frac{x}{57} d x} - 2 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=\frac{1}{57}$$$ と $$$f{\left(x \right)} = x$$$ に対して適用する:

$$- \frac{2 x^{3}}{3} + {\color{red}{\int{\frac{x}{57} d x}}} = - \frac{2 x^{3}}{3} + {\color{red}{\left(\frac{\int{x d x}}{57}\right)}}$$

$$$n=1$$$ 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:

$$- \frac{2 x^{3}}{3} + \frac{{\color{red}{\int{x d x}}}}{57}=- \frac{2 x^{3}}{3} + \frac{{\color{red}{\frac{x^{1 + 1}}{1 + 1}}}}{57}=- \frac{2 x^{3}}{3} + \frac{{\color{red}{\left(\frac{x^{2}}{2}\right)}}}{57}$$

したがって、

$$\int{\left(- 2 x^{2} + \frac{x}{57}\right)d x} = - \frac{2 x^{3}}{3} + \frac{x^{2}}{114}$$

簡単化せよ:

$$\int{\left(- 2 x^{2} + \frac{x}{57}\right)d x} = \frac{x^{2} \left(1 - 76 x\right)}{114}$$

積分定数を加える:

$$\int{\left(- 2 x^{2} + \frac{x}{57}\right)d x} = \frac{x^{2} \left(1 - 76 x\right)}{114}+C$$

$$$\int \left(- 2 x^{2} + \frac{x}{57}\right)\, dx = \frac{x^{2} \left(1 - 76 x\right)}{114} + C$$$A