Funktion $$$\coth{\left(x \right)}$$$ integraali

Määritä $$$\int \coth{\left(x \right)}\, dx$$$.

Esitä hyperbolinen kotangentti muodossa $$$\coth\left(x\right)=\frac{\cosh\left(x\right)}{\sinh\left(x\right)}$$$:

$${\color{red}{\int{\coth{\left(x \right)} d x}}} = {\color{red}{\int{\frac{\cosh{\left(x \right)}}{\sinh{\left(x \right)}} d x}}}$$

Olkoon $$$u=\sinh{\left(x \right)}$$$.

Tällöin $$$du=\left(\sinh{\left(x \right)}\right)^{\prime }dx = \cosh{\left(x \right)} dx$$$ (vaiheet ovat nähtävissä ») ja saamme, että $$$\cosh{\left(x \right)} dx = du$$$.

Siis,

$${\color{red}{\int{\frac{\cosh{\left(x \right)}}{\sinh{\left(x \right)}} d x}}} = {\color{red}{\int{\frac{1}{u} d u}}}$$

Funktion $$$\frac{1}{u}$$$ integraali on $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

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

Muista, että $$$u=\sinh{\left(x \right)}$$$:

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

Näin ollen,

$$\int{\coth{\left(x \right)} d x} = \ln{\left(\left|{\sinh{\left(x \right)}}\right| \right)}$$

Lisää integrointivakio:

$$\int{\coth{\left(x \right)} d x} = \ln{\left(\left|{\sinh{\left(x \right)}}\right| \right)}+C$$

$$$\int \coth{\left(x \right)}\, dx = \ln\left(\left|{\sinh{\left(x \right)}}\right|\right) + C$$$A