Integralen av $$$\frac{x}{\sqrt{- 6 x^{4} + 2 x^{2}}}$$$

Bestäm $$$\int \frac{x}{\sqrt{- 6 x^{4} + 2 x^{2}}}\, dx$$$.

Inmatningen skrivs om: $$$\int{\frac{x}{\sqrt{- 6 x^{4} + 2 x^{2}}} d x}=\int{\frac{1}{\sqrt{2 - 6 x^{2}}} d x}$$$.

Förenkla integranden:

$${\color{red}{\int{\frac{1}{\sqrt{2 - 6 x^{2}}} d x}}} = {\color{red}{\int{\frac{\sqrt{2}}{2 \sqrt{1 - 3 x^{2}}} d x}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{\sqrt{2}}{2}$$$ och $$$f{\left(x \right)} = \frac{1}{\sqrt{1 - 3 x^{2}}}$$$:

$${\color{red}{\int{\frac{\sqrt{2}}{2 \sqrt{1 - 3 x^{2}}} d x}}} = {\color{red}{\left(\frac{\sqrt{2} \int{\frac{1}{\sqrt{1 - 3 x^{2}}} d x}}{2}\right)}}$$

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

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

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

Alltså,

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

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

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

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

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

Integralen kan skrivas om som

$$\frac{\sqrt{2} {\color{red}{\int{\frac{1}{\sqrt{1 - 3 x^{2}}} d x}}}}{2} = \frac{\sqrt{2} {\color{red}{\int{\frac{\sqrt{3}}{3} d u}}}}{2}$$

Tillämpa konstantregeln $$$\int c\, du = c u$$$ med $$$c=\frac{\sqrt{3}}{3}$$$:

$$\frac{\sqrt{2} {\color{red}{\int{\frac{\sqrt{3}}{3} d u}}}}{2} = \frac{\sqrt{2} {\color{red}{\left(\frac{\sqrt{3} u}{3}\right)}}}{2}$$

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

$$\frac{\sqrt{6} {\color{red}{u}}}{6} = \frac{\sqrt{6} {\color{red}{\operatorname{asin}{\left(\sqrt{3} x \right)}}}}{6}$$

Alltså,

$$\int{\frac{1}{\sqrt{2 - 6 x^{2}}} d x} = \frac{\sqrt{6} \operatorname{asin}{\left(\sqrt{3} x \right)}}{6}$$

Lägg till integrationskonstanten:

$$\int{\frac{1}{\sqrt{2 - 6 x^{2}}} d x} = \frac{\sqrt{6} \operatorname{asin}{\left(\sqrt{3} x \right)}}{6}+C$$

$$$\int \frac{x}{\sqrt{- 6 x^{4} + 2 x^{2}}}\, dx = \frac{\sqrt{6} \operatorname{asin}{\left(\sqrt{3} x \right)}}{6} + C$$$A