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

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

Kvadratkomplettera (stegen kan ses »): $$$- x^{2} + x = \frac{1}{4} - \left(x - \frac{1}{2}\right)^{2}$$$:

$${\color{red}{\int{\frac{1}{\sqrt{- x^{2} + x}} d x}}} = {\color{red}{\int{\frac{1}{\sqrt{\frac{1}{4} - \left(x - \frac{1}{2}\right)^{2}}} d x}}}$$

Låt $$$u=x - \frac{1}{2}$$$ vara.

Då $$$du=\left(x - \frac{1}{2}\right)^{\prime }dx = 1 dx$$$ (stegen kan ses »), och vi har att $$$dx = du$$$.

Alltså,

$${\color{red}{\int{\frac{1}{\sqrt{\frac{1}{4} - \left(x - \frac{1}{2}\right)^{2}}} d x}}} = {\color{red}{\int{\frac{1}{\sqrt{\frac{1}{4} - u^{2}}} d u}}}$$

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

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

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

Alltså,

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

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

$$$\frac{1}{\sqrt{\frac{1}{4} - \frac{\sin^{2}{\left( v \right)}}{4}}}=\frac{2}{\sqrt{1 - \sin^{2}{\left( v \right)}}}=\frac{2}{\sqrt{\cos^{2}{\left( v \right)}}}$$$

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

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

Alltså,

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

Tillämpa konstantregeln $$$\int c\, dv = c v$$$ med $$$c=1$$$:

$${\color{red}{\int{1 d v}}} = {\color{red}{v}}$$

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

$${\color{red}{v}} = {\color{red}{\operatorname{asin}{\left(2 u \right)}}}$$

Kom ihåg att $$$u=x - \frac{1}{2}$$$:

$$\operatorname{asin}{\left(2 {\color{red}{u}} \right)} = \operatorname{asin}{\left(2 {\color{red}{\left(x - \frac{1}{2}\right)}} \right)}$$

Alltså,

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

Lägg till integrationskonstanten:

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

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