Ολοκλήρωμα του $$$\sin{\left(x \right)} - \cos{\left(x \right)}$$$

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

Ολοκληρώστε όρο προς όρο:

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

Το ολοκλήρωμα του συνημιτόνου είναι $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:

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

Το ολοκλήρωμα του ημιτόνου είναι $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:

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

Επομένως,

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

Απλοποιήστε:

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

Προσθέστε τη σταθερά ολοκλήρωσης:

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

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