Integralen av $$$- 5 \sec^{2}{\left(x \right)}$$$

Bestäm $$$\int \left(- 5 \sec^{2}{\left(x \right)}\right)\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=-5$$$ och $$$f{\left(x \right)} = \sec^{2}{\left(x \right)}$$$:

$${\color{red}{\int{\left(- 5 \sec^{2}{\left(x \right)}\right)d x}}} = {\color{red}{\left(- 5 \int{\sec^{2}{\left(x \right)} d x}\right)}}$$

Integralen av $$$\sec^{2}{\left(x \right)}$$$ är $$$\int{\sec^{2}{\left(x \right)} d x} = \tan{\left(x \right)}$$$:

$$- 5 {\color{red}{\int{\sec^{2}{\left(x \right)} d x}}} = - 5 {\color{red}{\tan{\left(x \right)}}}$$

Alltså,

$$\int{\left(- 5 \sec^{2}{\left(x \right)}\right)d x} = - 5 \tan{\left(x \right)}$$

Lägg till integrationskonstanten:

$$\int{\left(- 5 \sec^{2}{\left(x \right)}\right)d x} = - 5 \tan{\left(x \right)}+C$$

$$$\int \left(- 5 \sec^{2}{\left(x \right)}\right)\, dx = - 5 \tan{\left(x \right)} + C$$$A