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

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

Gliedweise integrieren:

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

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

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

Das Integral des Kosinus ist $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:

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

Daher,

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

Vereinfachen:

$$\int{\left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right)d x} = \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right)d x} = \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}+C$$

$$$\int \left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right)\, dx = \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)} + C$$$A