Integral of $$$\left(y - \sin{\left(y \right)}\right) \cos{\left(y \right)}$$$ with respect to $$$x$$$
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Your Input
Find $$$\int \left(y - \sin{\left(y \right)}\right) \cos{\left(y \right)}\, dx$$$.
Solution
Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=\left(y - \sin{\left(y \right)}\right) \cos{\left(y \right)}$$$:
$${\color{red}{\int{\left(y - \sin{\left(y \right)}\right) \cos{\left(y \right)} d x}}} = {\color{red}{x \left(y - \sin{\left(y \right)}\right) \cos{\left(y \right)}}}$$
Therefore,
$$\int{\left(y - \sin{\left(y \right)}\right) \cos{\left(y \right)} d x} = x \left(y - \sin{\left(y \right)}\right) \cos{\left(y \right)}$$
Add the constant of integration:
$$\int{\left(y - \sin{\left(y \right)}\right) \cos{\left(y \right)} d x} = x \left(y - \sin{\left(y \right)}\right) \cos{\left(y \right)}+C$$
Answer
$$$\int \left(y - \sin{\left(y \right)}\right) \cos{\left(y \right)}\, dx = x \left(y - \sin{\left(y \right)}\right) \cos{\left(y \right)} + C$$$A