Integraal van $$$\frac{1}{\sqrt{a - x^{2}}}$$$ met betrekking tot $$$x$$$

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

Zij $$$x=\sqrt{a} \sin{\left(u \right)}$$$.

Dan $$$dx=\left(\sqrt{a} \sin{\left(u \right)}\right)^{\prime }du = \sqrt{a} \cos{\left(u \right)} du$$$ (zie » voor de stappen).

Bovendien volgt dat $$$u=\operatorname{asin}{\left(\frac{x}{\sqrt{a}} \right)}$$$.

De integraand wordt

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

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

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

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

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

De integraal wordt

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

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 $$$u=\operatorname{asin}{\left(\frac{x}{\sqrt{a}} \right)}$$$:

$${\color{red}{u}} = {\color{red}{\operatorname{asin}{\left(\frac{x}{\sqrt{a}} \right)}}}$$

Dus,

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

Voeg de integratieconstante toe:

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

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