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

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

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

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

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

De integraand wordt

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

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

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

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

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

Dus,

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

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

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

Dus,

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

Voeg de integratieconstante toe:

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

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