Integral of $$$z - 10 \sin{\left(x \right)}$$$ with respect to $$$x$$$

Find $$$\int \left(z - 10 \sin{\left(x \right)}\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(z - 10 \sin{\left(x \right)}\right)d x}}} = {\color{red}{\left(\int{z d x} - \int{10 \sin{\left(x \right)} d x}\right)}}$$

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=z$$$:

$$- \int{10 \sin{\left(x \right)} d x} + {\color{red}{\int{z d x}}} = - \int{10 \sin{\left(x \right)} d x} + {\color{red}{x z}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=10$$$ and $$$f{\left(x \right)} = \sin{\left(x \right)}$$$:

$$x z - {\color{red}{\int{10 \sin{\left(x \right)} d x}}} = x z - {\color{red}{\left(10 \int{\sin{\left(x \right)} d x}\right)}}$$

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

$$x z - 10 {\color{red}{\int{\sin{\left(x \right)} d x}}} = x z - 10 {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$

Therefore,

$$\int{\left(z - 10 \sin{\left(x \right)}\right)d x} = x z + 10 \cos{\left(x \right)}$$

Add the constant of integration:

$$\int{\left(z - 10 \sin{\left(x \right)}\right)d x} = x z + 10 \cos{\left(x \right)}+C$$

$$$\int \left(z - 10 \sin{\left(x \right)}\right)\, dx = \left(x z + 10 \cos{\left(x \right)}\right) + C$$$A