Integraal van $$$\frac{1}{\sqrt{9 - x^{2}}}$$$

Bepaal $$$\int \frac{1}{\sqrt{9 - x^{2}}}\, dx$$$.

Zij $$$x=3 \sin{\left(u \right)}$$$.

Dan $$$dx=\left(3 \sin{\left(u \right)}\right)^{\prime }du = 3 \cos{\left(u \right)} du$$$ (zie » voor de stappen).

Bovendien volgt dat $$$u=\operatorname{asin}{\left(\frac{x}{3} \right)}$$$.

Dus,

$$$\frac{1}{\sqrt{9 - x^{2}}} = \frac{1}{\sqrt{9 - 9 \sin^{2}{\left( u \right)}}}$$$

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

$$$\frac{1}{\sqrt{9 - 9 \sin^{2}{\left( u \right)}}}=\frac{1}{3 \sqrt{1 - \sin^{2}{\left( u \right)}}}=\frac{1}{3 \sqrt{\cos^{2}{\left( u \right)}}}$$$

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

$$$\frac{1}{3 \sqrt{\cos^{2}{\left( u \right)}}} = \frac{1}{3 \cos{\left( u \right)}}$$$

Dus,

$${\color{red}{\int{\frac{1}{\sqrt{9 - x^{2}}} d x}}} = {\color{red}{\int{1 d u}}}$$

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 $$$u=\operatorname{asin}{\left(\frac{x}{3} \right)}$$$:

$${\color{red}{u}} = {\color{red}{\operatorname{asin}{\left(\frac{x}{3} \right)}}}$$

Dus,

$$\int{\frac{1}{\sqrt{9 - x^{2}}} d x} = \operatorname{asin}{\left(\frac{x}{3} \right)}$$

Voeg de integratieconstante toe:

$$\int{\frac{1}{\sqrt{9 - x^{2}}} d x} = \operatorname{asin}{\left(\frac{x}{3} \right)}+C$$

$$$\int \frac{1}{\sqrt{9 - x^{2}}}\, dx = \operatorname{asin}{\left(\frac{x}{3} \right)} + C$$$A