$$$y \sin{\left(x y \right)}$$$ 关于$$$x$$$的积分

求$$$\int y \sin{\left(x y \right)}\, dx$$$。

对 $$$c=y$$$ 和 $$$f{\left(x \right)} = \sin{\left(x y \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

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

设$$$u=x y$$$。

则$$$du=\left(x y\right)^{\prime }dx = y dx$$$ (步骤见»)，并有$$$dx = \frac{du}{y}$$$。

因此，

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

对 $$$c=\frac{1}{y}$$$ 和 $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

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

正弦函数的积分为 $$$\int{\sin{\left(u \right)} d u} = - \cos{\left(u \right)}$$$:

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

回忆一下 $$$u=x y$$$:

$$- \cos{\left({\color{red}{u}} \right)} = - \cos{\left({\color{red}{x y}} \right)}$$

因此，

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

加上积分常数：

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

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