Integral of $$$1 - \cos{\left(x \right)}$$$

Find $$$\int \left(1 - \cos{\left(x \right)}\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(1 - \cos{\left(x \right)}\right)d x}}} = {\color{red}{\left(\int{1 d x} - \int{\cos{\left(x \right)} d x}\right)}}$$

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=1$$$:

$$- \int{\cos{\left(x \right)} d x} + {\color{red}{\int{1 d x}}} = - \int{\cos{\left(x \right)} d x} + {\color{red}{x}}$$

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

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

Therefore,

$$\int{\left(1 - \cos{\left(x \right)}\right)d x} = x - \sin{\left(x \right)}$$

Add the constant of integration:

$$\int{\left(1 - \cos{\left(x \right)}\right)d x} = x - \sin{\left(x \right)}+C$$

$$$\int \left(1 - \cos{\left(x \right)}\right)\, dx = \left(x - \sin{\left(x \right)}\right) + C$$$A