Funktion $$$- \sin{\left(x \right)} + \cos{\left(x \right)}$$$ integraali

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

Integroi termi kerrallaan:

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

Sinifunktion integraali on $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:

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

Kosinin integraali on $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:

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

Näin ollen,

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

Sievennä:

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

Lisää integrointivakio:

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

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