Integraal van $$$\frac{1}{\sqrt{x^{2} - 4}}$$$

Bepaal $$$\int \frac{1}{\sqrt{x^{2} - 4}}\, dx$$$.

Zij $$$x=2 \cosh{\left(u \right)}$$$.

Dan $$$dx=\left(2 \cosh{\left(u \right)}\right)^{\prime }du = 2 \sinh{\left(u \right)} du$$$ (zie » voor de stappen).

Bovendien volgt dat $$$u=\operatorname{acosh}{\left(\frac{x}{2} \right)}$$$.

Dus,

$$$\frac{1}{\sqrt{x^{2} - 4}} = \frac{1}{\sqrt{4 \cosh^{2}{\left( u \right)} - 4}}$$$

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

$$$\frac{1}{\sqrt{4 \cosh^{2}{\left( u \right)} - 4}}=\frac{1}{2 \sqrt{\cosh^{2}{\left( u \right)} - 1}}=\frac{1}{2 \sqrt{\sinh^{2}{\left( u \right)}}}$$$

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

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

De integraal wordt

$${\color{red}{\int{\frac{1}{\sqrt{x^{2} - 4}} d x}}} = {\color{red}{\int{1 d u}}}$$

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 $$$u=\operatorname{acosh}{\left(\frac{x}{2} \right)}$$$:

$${\color{red}{u}} = {\color{red}{\operatorname{acosh}{\left(\frac{x}{2} \right)}}}$$

Dus,

$$\int{\frac{1}{\sqrt{x^{2} - 4}} d x} = \operatorname{acosh}{\left(\frac{x}{2} \right)}$$

Voeg de integratieconstante toe:

$$\int{\frac{1}{\sqrt{x^{2} - 4}} d x} = \operatorname{acosh}{\left(\frac{x}{2} \right)}+C$$

$$$\int \frac{1}{\sqrt{x^{2} - 4}}\, dx = \operatorname{acosh}{\left(\frac{x}{2} \right)} + C$$$A