$$$- x + \cos{\left(x \right)}$$$の積分

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

項別に積分せよ:

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

$$$n=1$$$ 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:

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

余弦の積分は$$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:

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

したがって、

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

積分定数を加える:

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

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