Integraal van $$$- \sin{\left(x \right)} + \cos{\left(x \right)}$$$
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Uw invoer
Bepaal $$$\int \left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right)\, dx$$$.
Oplossing
Integreer termgewijs:
$${\color{red}{\int{\left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right)d x}}} = {\color{red}{\left(- \int{\sin{\left(x \right)} d x} + \int{\cos{\left(x \right)} d x}\right)}}$$
De integraal van de sinus is $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:
$$\int{\cos{\left(x \right)} d x} - {\color{red}{\int{\sin{\left(x \right)} d x}}} = \int{\cos{\left(x \right)} d x} - {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$
De integraal van de cosinus is $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:
$$\cos{\left(x \right)} + {\color{red}{\int{\cos{\left(x \right)} d x}}} = \cos{\left(x \right)} + {\color{red}{\sin{\left(x \right)}}}$$
Dus,
$$\int{\left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right)d x} = \sin{\left(x \right)} + \cos{\left(x \right)}$$
Vereenvoudig:
$$\int{\left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right)d x} = \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}$$
Voeg de integratieconstante toe:
$$\int{\left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right)d x} = \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}+C$$
Antwoord
$$$\int \left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right)\, dx = \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)} + C$$$A