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

Bestäm $$$\int \cot{\left(\theta \right)}\, d\theta$$$.

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

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

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

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

Integralen kan omskrivas som

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

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 $$$u=\sin{\left(\theta \right)}$$$:

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

Alltså,

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

Lägg till integrationskonstanten:

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

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