$$$- 14 x + \sin{\left(x \right)}$$$の積分

$$$\int \left(- 14 x + \sin{\left(x \right)}\right)\, dx$$$ を求めよ。

項別に積分せよ:

$${\color{red}{\int{\left(- 14 x + \sin{\left(x \right)}\right)d x}}} = {\color{red}{\left(- \int{14 x d x} + \int{\sin{\left(x \right)} d x}\right)}}$$

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=14$$$ と $$$f{\left(x \right)} = x$$$ に対して適用する:

$$\int{\sin{\left(x \right)} d x} - {\color{red}{\int{14 x d x}}} = \int{\sin{\left(x \right)} d x} - {\color{red}{\left(14 \int{x d x}\right)}}$$

$$$n=1$$$ 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:

$$\int{\sin{\left(x \right)} d x} - 14 {\color{red}{\int{x d x}}}=\int{\sin{\left(x \right)} d x} - 14 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\int{\sin{\left(x \right)} d x} - 14 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

正弦関数の不定積分は$$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$です:

$$- 7 x^{2} + {\color{red}{\int{\sin{\left(x \right)} d x}}} = - 7 x^{2} + {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$

したがって、

$$\int{\left(- 14 x + \sin{\left(x \right)}\right)d x} = - 7 x^{2} - \cos{\left(x \right)}$$

積分定数を加える:

$$\int{\left(- 14 x + \sin{\left(x \right)}\right)d x} = - 7 x^{2} - \cos{\left(x \right)}+C$$

$$$\int \left(- 14 x + \sin{\left(x \right)}\right)\, dx = \left(- 7 x^{2} - \cos{\left(x \right)}\right) + C$$$A