Integral of $$$y \sin{\left(x y \right)}$$$ with respect to $$$x$$$

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=y$$$ and $$$f{\left(x \right)} = \sin{\left(x y \right)}$$$:

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

Let $$$u=x y$$$.

Then $$$du=\left(x y\right)^{\prime }dx = y dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{y}$$$.

The integral can be rewritten as

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

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{y}$$$ and $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

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

The integral of the sine is $$$\int{\sin{\left(u \right)} d u} = - \cos{\left(u \right)}$$$:

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

Recall that $$$u=x y$$$:

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

Therefore,

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

Add the constant of integration:

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

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