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Integraal van $$$\cot{\left(\pi x \right)}$$$
Gerelateerde rekenmachine: Rekenmachine voor bepaalde en oneigenlijke integralen
Uw invoer
Bepaal $$$\int \cot{\left(\pi x \right)}\, dx$$$.
Oplossing
Zij $$$u=\pi x$$$.
Dan $$$du=\left(\pi x\right)^{\prime }dx = \pi dx$$$ (de stappen zijn te zien »), en dan geldt dat $$$dx = \frac{du}{\pi}$$$.
De integraal wordt
$${\color{red}{\int{\cot{\left(\pi x \right)} d x}}} = {\color{red}{\int{\frac{\cot{\left(u \right)}}{\pi} d u}}}$$
Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=\frac{1}{\pi}$$$ en $$$f{\left(u \right)} = \cot{\left(u \right)}$$$:
$${\color{red}{\int{\frac{\cot{\left(u \right)}}{\pi} d u}}} = {\color{red}{\frac{\int{\cot{\left(u \right)} d u}}{\pi}}}$$
Herschrijf de cotangens als $$$\cot\left( u \right)=\frac{\cos\left( u \right)}{\sin\left( u \right)}$$$:
$$\frac{{\color{red}{\int{\cot{\left(u \right)} d u}}}}{\pi} = \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{\sin{\left(u \right)}} d u}}}}{\pi}$$
Zij $$$v=\sin{\left(u \right)}$$$.
Dan $$$dv=\left(\sin{\left(u \right)}\right)^{\prime }du = \cos{\left(u \right)} du$$$ (de stappen zijn te zien »), en dan geldt dat $$$\cos{\left(u \right)} du = dv$$$.
De integraal wordt
$$\frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{\sin{\left(u \right)}} d u}}}}{\pi} = \frac{{\color{red}{\int{\frac{1}{v} d v}}}}{\pi}$$
De integraal van $$$\frac{1}{v}$$$ is $$$\int{\frac{1}{v} d v} = \ln{\left(\left|{v}\right| \right)}$$$:
$$\frac{{\color{red}{\int{\frac{1}{v} d v}}}}{\pi} = \frac{{\color{red}{\ln{\left(\left|{v}\right| \right)}}}}{\pi}$$
We herinneren eraan dat $$$v=\sin{\left(u \right)}$$$:
$$\frac{\ln{\left(\left|{{\color{red}{v}}}\right| \right)}}{\pi} = \frac{\ln{\left(\left|{{\color{red}{\sin{\left(u \right)}}}}\right| \right)}}{\pi}$$
We herinneren eraan dat $$$u=\pi x$$$:
$$\frac{\ln{\left(\left|{\sin{\left({\color{red}{u}} \right)}}\right| \right)}}{\pi} = \frac{\ln{\left(\left|{\sin{\left({\color{red}{\pi x}} \right)}}\right| \right)}}{\pi}$$
Dus,
$$\int{\cot{\left(\pi x \right)} d x} = \frac{\ln{\left(\left|{\sin{\left(\pi x \right)}}\right| \right)}}{\pi}$$
Voeg de integratieconstante toe:
$$\int{\cot{\left(\pi x \right)} d x} = \frac{\ln{\left(\left|{\sin{\left(\pi x \right)}}\right| \right)}}{\pi}+C$$
Antwoord
$$$\int \cot{\left(\pi x \right)}\, dx = \frac{\ln\left(\left|{\sin{\left(\pi x \right)}}\right|\right)}{\pi} + C$$$A