Integralen av $$$\frac{1}{\sqrt{9 - 4 x^{2}}}$$$

Bestäm $$$\int \frac{1}{\sqrt{9 - 4 x^{2}}}\, dx$$$.

Låt $$$x=\frac{3 \sin{\left(u \right)}}{2}$$$ vara.

Då $$$dx=\left(\frac{3 \sin{\left(u \right)}}{2}\right)^{\prime }du = \frac{3 \cos{\left(u \right)}}{2} du$$$ (stegen kan ses »).

Det följer också att $$$u=\operatorname{asin}{\left(\frac{2 x}{3} \right)}$$$.

Alltså,

$$$\frac{1}{\sqrt{9 - 4 x^{2}}} = \frac{1}{\sqrt{9 - 9 \sin^{2}{\left( u \right)}}}$$$

Använd identiteten $$$1 - \sin^{2}{\left( u \right)} = \cos^{2}{\left( u \right)}$$$:

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

Om vi antar att $$$\cos{\left( u \right)} \ge 0$$$, erhåller vi följande:

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

Alltså,

$${\color{red}{\int{\frac{1}{\sqrt{9 - 4 x^{2}}} d x}}} = {\color{red}{\int{\frac{1}{2} d u}}}$$

Tillämpa konstantregeln $$$\int c\, du = c u$$$ med $$$c=\frac{1}{2}$$$:

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

Kom ihåg att $$$u=\operatorname{asin}{\left(\frac{2 x}{3} \right)}$$$:

$$\frac{{\color{red}{u}}}{2} = \frac{{\color{red}{\operatorname{asin}{\left(\frac{2 x}{3} \right)}}}}{2}$$

Alltså,

$$\int{\frac{1}{\sqrt{9 - 4 x^{2}}} d x} = \frac{\operatorname{asin}{\left(\frac{2 x}{3} \right)}}{2}$$

Lägg till integrationskonstanten:

$$\int{\frac{1}{\sqrt{9 - 4 x^{2}}} d x} = \frac{\operatorname{asin}{\left(\frac{2 x}{3} \right)}}{2}+C$$

$$$\int \frac{1}{\sqrt{9 - 4 x^{2}}}\, dx = \frac{\operatorname{asin}{\left(\frac{2 x}{3} \right)}}{2} + C$$$A