Integral von $$$1 - \sin{\left(x \right)}$$$

Bestimme $$$\int \left(1 - \sin{\left(x \right)}\right)\, dx$$$.

Gliedweise integrieren:

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

Wenden Sie die Konstantenregel $$$\int c\, dx = c x$$$ mit $$$c=1$$$ an:

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

Das Integral des Sinus lautet $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:

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

Daher,

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

Fügen Sie die Integrationskonstante hinzu:

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

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