Integraal van $$$\frac{\sqrt{x^{2} - 1}}{x - 1}$$$

Bepaal $$$\int \frac{\sqrt{x^{2} - 1}}{x - 1}\, dx$$$.

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

Dan $$$dx=\left(\cosh{\left(u \right)}\right)^{\prime }du = \sinh{\left(u \right)} du$$$ (zie » voor de stappen).

Bovendien volgt dat $$$u=\operatorname{acosh}{\left(x \right)}$$$.

Dus,

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

Gebruik de identiteit $$$\cosh^{2}{\left( u \right)} - 1 = \sinh^{2}{\left( u \right)}$$$:

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

Aangenomen dat $$$\sinh{\left( u \right)} \ge 0$$$, verkrijgen we het volgende:

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

Dus,

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

Herschrijf de hyperbolische sinus in termen van de hyperbolische cosinus, werk de teller verder uit, gebruik de formule voor het verschil van kwadraten en vereenvoudig:

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

Integreer termgewijs:

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

Pas de constantenregel $$$\int c\, du = c u$$$ toe met $$$c=1$$$:

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

De integraal van de cosinus hyperbolicus is $$$\int{\cosh{\left(u \right)} d u} = \sinh{\left(u \right)}$$$:

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

We herinneren eraan dat $$$u=\operatorname{acosh}{\left(x \right)}$$$:

$$\sinh{\left({\color{red}{u}} \right)} + {\color{red}{u}} = \sinh{\left({\color{red}{\operatorname{acosh}{\left(x \right)}}} \right)} + {\color{red}{\operatorname{acosh}{\left(x \right)}}}$$

Dus,

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

Voeg de integratieconstante toe:

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

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