Integralen av $$$- \cot{\left(x \right)}$$$

Bestäm $$$\int \left(- \cot{\left(x \right)}\right)\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=-1$$$ och $$$f{\left(x \right)} = \cot{\left(x \right)}$$$:

$${\color{red}{\int{\left(- \cot{\left(x \right)}\right)d x}}} = {\color{red}{\left(- \int{\cot{\left(x \right)} d x}\right)}}$$

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

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

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

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

Alltså,

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

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

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

Alltså,

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

Lägg till integrationskonstanten:

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

$$$\int \left(- \cot{\left(x \right)}\right)\, dx = - \ln\left(\left|{\sin{\left(x \right)}}\right|\right) + C$$$A