Integralen av $$$- \frac{1}{\sqrt{a^{2} - x^{2}}}$$$ med avseende på $$$x$$$

Bestäm $$$\int \left(- \frac{1}{\sqrt{a^{2} - x^{2}}}\right)\, dx$$$.

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

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

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

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

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

Alltså,

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

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

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

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

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

Alltså,

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

Tillämpa konstantregeln $$$\int c\, du = c u$$$ med $$$c=1$$$:

$$- {\color{red}{\int{1 d u}}} = - {\color{red}{u}}$$

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

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

Alltså,

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

Lägg till integrationskonstanten:

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

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