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

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

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=2$$$ en $$$f{\left(x \right)} = \frac{1}{x \sqrt{x^{2} - 4}}$$$:

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

Zij $$$u=\frac{1}{x}$$$.

Dan $$$du=\left(\frac{1}{x}\right)^{\prime }dx = - \frac{1}{x^{2}} dx$$$ (de stappen zijn te zien »), en dan geldt dat $$$\frac{dx}{x^{2}} = - du$$$.

De integraal kan worden herschreven als

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

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=-1$$$ en $$$f{\left(u \right)} = \frac{1}{\sqrt{1 - 4 u^{2}}}$$$:

$$2 {\color{red}{\int{\left(- \frac{1}{\sqrt{1 - 4 u^{2}}}\right)d u}}} = 2 {\color{red}{\left(- \int{\frac{1}{\sqrt{1 - 4 u^{2}}} d u}\right)}}$$

Zij $$$u=\frac{\sin{\left(v \right)}}{2}$$$.

Dan $$$du=\left(\frac{\sin{\left(v \right)}}{2}\right)^{\prime }dv = \frac{\cos{\left(v \right)}}{2} dv$$$ (zie » voor de stappen).

Bovendien volgt dat $$$v=\operatorname{asin}{\left(2 u \right)}$$$.

Dus,

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

Gebruik de identiteit $$$1 - \sin^{2}{\left( v \right)} = \cos^{2}{\left( v \right)}$$$:

$$$\frac{1}{\sqrt{1 - \sin^{2}{\left( v \right)}}}=\frac{1}{\sqrt{\cos^{2}{\left( v \right)}}}$$$

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

$$$\frac{1}{\sqrt{\cos^{2}{\left( v \right)}}} = \frac{1}{\cos{\left( v \right)}}$$$

Dus,

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

Pas de constantenregel $$$\int c\, dv = c v$$$ toe met $$$c=\frac{1}{2}$$$:

$$- 2 {\color{red}{\int{\frac{1}{2} d v}}} = - 2 {\color{red}{\left(\frac{v}{2}\right)}}$$

We herinneren eraan dat $$$v=\operatorname{asin}{\left(2 u \right)}$$$:

$$- {\color{red}{v}} = - {\color{red}{\operatorname{asin}{\left(2 u \right)}}}$$

We herinneren eraan dat $$$u=\frac{1}{x}$$$:

$$- \operatorname{asin}{\left(2 {\color{red}{u}} \right)} = - \operatorname{asin}{\left(2 {\color{red}{\frac{1}{x}}} \right)}$$

Dus,

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

Voeg de integratieconstante toe:

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

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