Integralen av $$$\tan^{3}{\left(x \right)} \cot^{3}{\left(x \right)}$$$

Bestäm $$$\int \tan^{3}{\left(x \right)} \cot^{3}{\left(x \right)}\, dx$$$.

Förenkla integranden:

$${\color{red}{\int{\tan^{3}{\left(x \right)} \cot^{3}{\left(x \right)} d x}}} = {\color{red}{\int{1 d x}}}$$

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=1$$$:

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

Alltså,

$$\int{\tan^{3}{\left(x \right)} \cot^{3}{\left(x \right)} d x} = x$$

Lägg till integrationskonstanten:

$$\int{\tan^{3}{\left(x \right)} \cot^{3}{\left(x \right)} d x} = x+C$$

$$$\int \tan^{3}{\left(x \right)} \cot^{3}{\left(x \right)}\, dx = x + C$$$A