Integralen av $$$\frac{1}{\left(x^{2} - 1\right)^{\frac{3}{2}}}$$$

Bestäm $$$\int \frac{1}{\left(x^{2} - 1\right)^{\frac{3}{2}}}\, dx$$$.

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

Då $$$dx=\left(\cosh{\left(u \right)}\right)^{\prime }du = \sinh{\left(u \right)} du$$$ (stegen kan ses »).

Det följer också att $$$u=\operatorname{acosh}{\left(x \right)}$$$.

Alltså,

$$$\frac{1}{\left(x^{2} - 1\right)^{\frac{3}{2}}} = \frac{1}{\left(\cosh^{2}{\left( u \right)} - 1\right)^{\frac{3}{2}}}$$$

Använd identiteten $$$\cosh^{2}{\left( u \right)} - 1 = \sinh^{2}{\left( u \right)}$$$:

$$$\frac{1}{\left(\cosh^{2}{\left( u \right)} - 1\right)^{\frac{3}{2}}}=\frac{1}{\left(\sinh^{2}{\left( u \right)}\right)^{\frac{3}{2}}}$$$

Om vi antar att $$$\sinh{\left( u \right)} \ge 0$$$, erhåller vi följande:

$$$\frac{1}{\left(\sinh^{2}{\left( u \right)}\right)^{\frac{3}{2}}} = \frac{1}{\sinh^{3}{\left( u \right)}}$$$

Alltså,

$${\color{red}{\int{\frac{1}{\left(x^{2} - 1\right)^{\frac{3}{2}}} d x}}} = {\color{red}{\int{\frac{1}{\sinh^{2}{\left(u \right)}} d u}}}$$

Skriv om integranden i termer av den hyperboliska kosekanten:

$${\color{red}{\int{\frac{1}{\sinh^{2}{\left(u \right)}} d u}}} = {\color{red}{\int{\operatorname{csch}^{2}{\left(u \right)} d u}}}$$

Integralen av $$$\operatorname{csch}^{2}{\left(u \right)}$$$ är $$$\int{\operatorname{csch}^{2}{\left(u \right)} d u} = - \coth{\left(u \right)}$$$:

$${\color{red}{\int{\operatorname{csch}^{2}{\left(u \right)} d u}}} = {\color{red}{\left(- \coth{\left(u \right)}\right)}}$$

Kom ihåg att $$$u=\operatorname{acosh}{\left(x \right)}$$$:

$$- \coth{\left({\color{red}{u}} \right)} = - \coth{\left({\color{red}{\operatorname{acosh}{\left(x \right)}}} \right)}$$

Alltså,

$$\int{\frac{1}{\left(x^{2} - 1\right)^{\frac{3}{2}}} d x} = - \frac{x}{\sqrt{x - 1} \sqrt{x + 1}}$$

Lägg till integrationskonstanten:

$$\int{\frac{1}{\left(x^{2} - 1\right)^{\frac{3}{2}}} d x} = - \frac{x}{\sqrt{x - 1} \sqrt{x + 1}}+C$$

$$$\int \frac{1}{\left(x^{2} - 1\right)^{\frac{3}{2}}}\, dx = - \frac{x}{\sqrt{x - 1} \sqrt{x + 1}} + C$$$A