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Integraal van $$$1 - \sec^{2}{\left(x \right)}$$$
Gerelateerde rekenmachine: Rekenmachine voor bepaalde en oneigenlijke integralen
Uw invoer
Bepaal $$$\int \left(1 - \sec^{2}{\left(x \right)}\right)\, dx$$$.
Oplossing
Integreer termgewijs:
$${\color{red}{\int{\left(1 - \sec^{2}{\left(x \right)}\right)d x}}} = {\color{red}{\left(\int{1 d x} - \int{\sec^{2}{\left(x \right)} d x}\right)}}$$
Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=1$$$:
$$- \int{\sec^{2}{\left(x \right)} d x} + {\color{red}{\int{1 d x}}} = - \int{\sec^{2}{\left(x \right)} d x} + {\color{red}{x}}$$
De integraal van $$$\sec^{2}{\left(x \right)}$$$ is $$$\int{\sec^{2}{\left(x \right)} d x} = \tan{\left(x \right)}$$$:
$$x - {\color{red}{\int{\sec^{2}{\left(x \right)} d x}}} = x - {\color{red}{\tan{\left(x \right)}}}$$
Dus,
$$\int{\left(1 - \sec^{2}{\left(x \right)}\right)d x} = x - \tan{\left(x \right)}$$
Voeg de integratieconstante toe:
$$\int{\left(1 - \sec^{2}{\left(x \right)}\right)d x} = x - \tan{\left(x \right)}+C$$
Antwoord
$$$\int \left(1 - \sec^{2}{\left(x \right)}\right)\, dx = \left(x - \tan{\left(x \right)}\right) + C$$$A