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Integraal van $$$\sin{\left(x \right)} - \cos{\left(x \right)}$$$
Gerelateerde rekenmachine: Rekenmachine voor bepaalde en oneigenlijke integralen
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 cosinus is $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:
$$\int{\sin{\left(x \right)} d x} - {\color{red}{\int{\cos{\left(x \right)} d x}}} = \int{\sin{\left(x \right)} d x} - {\color{red}{\sin{\left(x \right)}}}$$
De integraal van de sinus is $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:
$$- \sin{\left(x \right)} + {\color{red}{\int{\sin{\left(x \right)} d x}}} = - \sin{\left(x \right)} + {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$
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