Integralen av $$$\frac{\tan{\left(x \right)}}{\cos{\left(x \right)}}$$$

Bestäm $$$\int \frac{\tan{\left(x \right)}}{\cos{\left(x \right)}}\, dx$$$.

Skriv om integranden:

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

Låt $$$u=\sec{\left(x \right)}$$$ vara.

Då $$$du=\left(\sec{\left(x \right)}\right)^{\prime }dx = \tan{\left(x \right)} \sec{\left(x \right)} dx$$$ (stegen kan ses »), och vi har att $$$\tan{\left(x \right)} \sec{\left(x \right)} dx = du$$$.

Alltså,

$${\color{red}{\int{\tan{\left(x \right)} \sec{\left(x \right)} d x}}} = {\color{red}{\int{1 d u}}}$$

Tillämpa konstantregeln $$$\int c\, du = c u$$$ med $$$c=1$$$:

$${\color{red}{\int{1 d u}}} = {\color{red}{u}}$$

Kom ihåg att $$$u=\sec{\left(x \right)}$$$:

$${\color{red}{u}} = {\color{red}{\sec{\left(x \right)}}}$$

Alltså,

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

Lägg till integrationskonstanten:

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

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