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