Integralen av $$$\cot{\left(v \right)}$$$

Bestäm $$$\int \cot{\left(v \right)}\, dv$$$.

Skriv om kotangensen som $$$\cot\left(v\right)=\frac{\cos\left(v\right)}{\sin\left(v\right)}$$$:

$${\color{red}{\int{\cot{\left(v \right)} d v}}} = {\color{red}{\int{\frac{\cos{\left(v \right)}}{\sin{\left(v \right)}} d v}}}$$

Låt $$$u=\sin{\left(v \right)}$$$ vara.

Då $$$du=\left(\sin{\left(v \right)}\right)^{\prime }dv = \cos{\left(v \right)} dv$$$ (stegen kan ses »), och vi har att $$$\cos{\left(v \right)} dv = du$$$.

Integralen blir

$${\color{red}{\int{\frac{\cos{\left(v \right)}}{\sin{\left(v \right)}} d v}}} = {\color{red}{\int{\frac{1}{u} d u}}}$$

Integralen av $$$\frac{1}{u}$$$ är $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

$${\color{red}{\int{\frac{1}{u} d u}}} = {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

Kom ihåg att $$$u=\sin{\left(v \right)}$$$:

$$\ln{\left(\left|{{\color{red}{u}}}\right| \right)} = \ln{\left(\left|{{\color{red}{\sin{\left(v \right)}}}}\right| \right)}$$

Alltså,

$$\int{\cot{\left(v \right)} d v} = \ln{\left(\left|{\sin{\left(v \right)}}\right| \right)}$$

Lägg till integrationskonstanten:

$$\int{\cot{\left(v \right)} d v} = \ln{\left(\left|{\sin{\left(v \right)}}\right| \right)}+C$$

$$$\int \cot{\left(v \right)}\, dv = \ln\left(\left|{\sin{\left(v \right)}}\right|\right) + C$$$A