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

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

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

De integraand wordt

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

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

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

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

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

Dus,

$${\color{red}{\int{\frac{1}{\sqrt{x^{2} - 1}} d x}}} = {\color{red}{\int{1 d u}}}$$

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

$${\color{red}{\int{1 d u}}} = {\color{red}{u}}$$

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

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

Dus,

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

Voeg de integratieconstante toe:

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

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