Integralen av $$$\cot{\left(22 x \right)}$$$

Bestäm $$$\int \cot{\left(22 x \right)}\, dx$$$.

Låt $$$u=22 x$$$ vara.

Då $$$du=\left(22 x\right)^{\prime }dx = 22 dx$$$ (stegen kan ses »), och vi har att $$$dx = \frac{du}{22}$$$.

Integralen blir

$${\color{red}{\int{\cot{\left(22 x \right)} d x}}} = {\color{red}{\int{\frac{\cot{\left(u \right)}}{22} d u}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=\frac{1}{22}$$$ och $$$f{\left(u \right)} = \cot{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\cot{\left(u \right)}}{22} d u}}} = {\color{red}{\left(\frac{\int{\cot{\left(u \right)} d u}}{22}\right)}}$$

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

$$\frac{{\color{red}{\int{\cot{\left(u \right)} d u}}}}{22} = \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{\sin{\left(u \right)}} d u}}}}{22}$$

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

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

Alltså,

$$\frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{\sin{\left(u \right)}} d u}}}}{22} = \frac{{\color{red}{\int{\frac{1}{v} d v}}}}{22}$$

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

$$\frac{{\color{red}{\int{\frac{1}{v} d v}}}}{22} = \frac{{\color{red}{\ln{\left(\left|{v}\right| \right)}}}}{22}$$

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

$$\frac{\ln{\left(\left|{{\color{red}{v}}}\right| \right)}}{22} = \frac{\ln{\left(\left|{{\color{red}{\sin{\left(u \right)}}}}\right| \right)}}{22}$$

Kom ihåg att $$$u=22 x$$$:

$$\frac{\ln{\left(\left|{\sin{\left({\color{red}{u}} \right)}}\right| \right)}}{22} = \frac{\ln{\left(\left|{\sin{\left({\color{red}{\left(22 x\right)}} \right)}}\right| \right)}}{22}$$

Alltså,

$$\int{\cot{\left(22 x \right)} d x} = \frac{\ln{\left(\left|{\sin{\left(22 x \right)}}\right| \right)}}{22}$$

Lägg till integrationskonstanten:

$$\int{\cot{\left(22 x \right)} d x} = \frac{\ln{\left(\left|{\sin{\left(22 x \right)}}\right| \right)}}{22}+C$$

$$$\int \cot{\left(22 x \right)}\, dx = \frac{\ln\left(\left|{\sin{\left(22 x \right)}}\right|\right)}{22} + C$$$A