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

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

Integre termo a termo:

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

A integral do cosseno é $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:

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

A integral do seno é $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:

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

Portanto,

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

Simplifique:

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

Adicione a constante de integração:

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

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