Integral de $$$z - 10 \sin{\left(x \right)}$$$ con respecto a $$$x$$$

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

Integra término a término:

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

Aplica la regla de la constante $$$\int c\, dx = c x$$$ con $$$c=z$$$:

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

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=10$$$ y $$$f{\left(x \right)} = \sin{\left(x \right)}$$$:

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

La integral del seno es $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:

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

Por lo tanto,

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

Añade la constante de integración:

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

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