Funktion $$$\frac{\sec{\left(x \right)}}{\cos{\left(x \right)}}$$$ integraali

Määritä $$$\int \frac{\sec{\left(x \right)}}{\cos{\left(x \right)}}\, dx$$$.

Kirjoita integroituva uudelleen:

$${\color{red}{\int{\frac{\sec{\left(x \right)}}{\cos{\left(x \right)}} d x}}} = {\color{red}{\int{\sec^{2}{\left(x \right)} d x}}}$$

Funktion $$$\sec^{2}{\left(x \right)}$$$ integraali on $$$\int{\sec^{2}{\left(x \right)} d x} = \tan{\left(x \right)}$$$:

$${\color{red}{\int{\sec^{2}{\left(x \right)} d x}}} = {\color{red}{\tan{\left(x \right)}}}$$

Näin ollen,

$$\int{\frac{\sec{\left(x \right)}}{\cos{\left(x \right)}} d x} = \tan{\left(x \right)}$$

Lisää integrointivakio:

$$\int{\frac{\sec{\left(x \right)}}{\cos{\left(x \right)}} d x} = \tan{\left(x \right)}+C$$

$$$\int \frac{\sec{\left(x \right)}}{\cos{\left(x \right)}}\, dx = \tan{\left(x \right)} + C$$$A