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

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

Låt $$$u=\pi x$$$ vara.

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

Integralen blir

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

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

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

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

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

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 $$$\cos{\left(u \right)} du = dv$$$.

Integralen kan omskrivas som

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

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

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

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

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

Kom ihåg att $$$u=\pi x$$$:

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

Alltså,

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

Lägg till integrationskonstanten:

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

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