$$$\frac{\sin{\left(x \right)}}{y}$$$ の $$$x$$$ に関する積分

$$$\int \frac{\sin{\left(x \right)}}{y}\, dx$$$ を求めよ。

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

$${\color{red}{\int{\frac{\sin{\left(x \right)}}{y} d x}}} = {\color{red}{\frac{\int{\sin{\left(x \right)} d x}}{y}}}$$

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

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

したがって、

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

積分定数を加える:

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

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