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Integraal van $$$\frac{1}{\cos^{2}{\left(\theta \right)}}$$$
Gerelateerde rekenmachine: Rekenmachine voor bepaalde en oneigenlijke integralen
Uw invoer
Bepaal $$$\int \frac{1}{\cos^{2}{\left(\theta \right)}}\, d\theta$$$.
Oplossing
Herschrijf de integraand in termen van secans:
$${\color{red}{\int{\frac{1}{\cos^{2}{\left(\theta \right)}} d \theta}}} = {\color{red}{\int{\sec^{2}{\left(\theta \right)} d \theta}}}$$
De integraal van $$$\sec^{2}{\left(\theta \right)}$$$ is $$$\int{\sec^{2}{\left(\theta \right)} d \theta} = \tan{\left(\theta \right)}$$$:
$${\color{red}{\int{\sec^{2}{\left(\theta \right)} d \theta}}} = {\color{red}{\tan{\left(\theta \right)}}}$$
Dus,
$$\int{\frac{1}{\cos^{2}{\left(\theta \right)}} d \theta} = \tan{\left(\theta \right)}$$
Voeg de integratieconstante toe:
$$\int{\frac{1}{\cos^{2}{\left(\theta \right)}} d \theta} = \tan{\left(\theta \right)}+C$$
Antwoord
$$$\int \frac{1}{\cos^{2}{\left(\theta \right)}}\, d\theta = \tan{\left(\theta \right)} + C$$$A