Integral dari $$$y \sin{\left(x y \right)}$$$ terhadap $$$x$$$

Temukan $$$\int y \sin{\left(x y \right)}\, dx$$$.

Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=y$$$ dan $$$f{\left(x \right)} = \sin{\left(x y \right)}$$$:

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

Misalkan $$$u=x y$$$.

Kemudian $$$du=\left(x y\right)^{\prime }dx = y dx$$$ (langkah-langkah dapat dilihat di »), dan kita memperoleh $$$dx = \frac{du}{y}$$$.

Oleh karena itu,

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

Terapkan aturan pengali konstanta $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ dengan $$$c=\frac{1}{y}$$$ dan $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

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

Integral dari sinus adalah $$$\int{\sin{\left(u \right)} d u} = - \cos{\left(u \right)}$$$:

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

Ingat bahwa $$$u=x y$$$:

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

Oleh karena itu,

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

Tambahkan konstanta integrasi:

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

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