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

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

Kvadratkomplettera (stegen kan ses »): $$$- x^{2} + 6 x = 9 - \left(x - 3\right)^{2}$$$:

$${\color{red}{\int{\sqrt{- x^{2} + 6 x} d x}}} = {\color{red}{\int{\sqrt{9 - \left(x - 3\right)^{2}} d x}}}$$

Låt $$$u=x - 3$$$ vara.

Då $$$du=\left(x - 3\right)^{\prime }dx = 1 dx$$$ (stegen kan ses »), och vi har att $$$dx = du$$$.

Alltså,

$${\color{red}{\int{\sqrt{9 - \left(x - 3\right)^{2}} d x}}} = {\color{red}{\int{\sqrt{9 - u^{2}} d u}}}$$

Låt $$$u=3 \sin{\left(v \right)}$$$ vara.

Då $$$du=\left(3 \sin{\left(v \right)}\right)^{\prime }dv = 3 \cos{\left(v \right)} dv$$$ (stegen kan ses »).

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

Alltså,

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

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

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

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

$$$3 \sqrt{\cos^{2}{\left( v \right)}} = 3 \cos{\left( v \right)}$$$

Alltså,

$${\color{red}{\int{\sqrt{9 - u^{2}} d u}}} = {\color{red}{\int{9 \cos^{2}{\left(v \right)} d v}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ med $$$c=9$$$ och $$$f{\left(v \right)} = \cos^{2}{\left(v \right)}$$$:

$${\color{red}{\int{9 \cos^{2}{\left(v \right)} d v}}} = {\color{red}{\left(9 \int{\cos^{2}{\left(v \right)} d v}\right)}}$$

Använd potensreduceringsformeln $$$\cos^{2}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{1}{2}$$$ med $$$\alpha= v $$$:

$$9 {\color{red}{\int{\cos^{2}{\left(v \right)} d v}}} = 9 {\color{red}{\int{\left(\frac{\cos{\left(2 v \right)}}{2} + \frac{1}{2}\right)d v}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ med $$$c=\frac{1}{2}$$$ och $$$f{\left(v \right)} = \cos{\left(2 v \right)} + 1$$$:

$$9 {\color{red}{\int{\left(\frac{\cos{\left(2 v \right)}}{2} + \frac{1}{2}\right)d v}}} = 9 {\color{red}{\left(\frac{\int{\left(\cos{\left(2 v \right)} + 1\right)d v}}{2}\right)}}$$

Integrera termvis:

$$\frac{9 {\color{red}{\int{\left(\cos{\left(2 v \right)} + 1\right)d v}}}}{2} = \frac{9 {\color{red}{\left(\int{1 d v} + \int{\cos{\left(2 v \right)} d v}\right)}}}{2}$$

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

$$\frac{9 \int{\cos{\left(2 v \right)} d v}}{2} + \frac{9 {\color{red}{\int{1 d v}}}}{2} = \frac{9 \int{\cos{\left(2 v \right)} d v}}{2} + \frac{9 {\color{red}{v}}}{2}$$

Låt $$$w=2 v$$$ vara.

Då $$$dw=\left(2 v\right)^{\prime }dv = 2 dv$$$ (stegen kan ses »), och vi har att $$$dv = \frac{dw}{2}$$$.

Integralen blir

$$\frac{9 v}{2} + \frac{9 {\color{red}{\int{\cos{\left(2 v \right)} d v}}}}{2} = \frac{9 v}{2} + \frac{9 {\color{red}{\int{\frac{\cos{\left(w \right)}}{2} d w}}}}{2}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(w \right)}\, dw = c \int f{\left(w \right)}\, dw$$$ med $$$c=\frac{1}{2}$$$ och $$$f{\left(w \right)} = \cos{\left(w \right)}$$$:

$$\frac{9 v}{2} + \frac{9 {\color{red}{\int{\frac{\cos{\left(w \right)}}{2} d w}}}}{2} = \frac{9 v}{2} + \frac{9 {\color{red}{\left(\frac{\int{\cos{\left(w \right)} d w}}{2}\right)}}}{2}$$

Integralen av cosinus är $$$\int{\cos{\left(w \right)} d w} = \sin{\left(w \right)}$$$:

$$\frac{9 v}{2} + \frac{9 {\color{red}{\int{\cos{\left(w \right)} d w}}}}{4} = \frac{9 v}{2} + \frac{9 {\color{red}{\sin{\left(w \right)}}}}{4}$$

Kom ihåg att $$$w=2 v$$$:

$$\frac{9 v}{2} + \frac{9 \sin{\left({\color{red}{w}} \right)}}{4} = \frac{9 v}{2} + \frac{9 \sin{\left({\color{red}{\left(2 v\right)}} \right)}}{4}$$

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

$$\frac{9 \sin{\left(2 {\color{red}{v}} \right)}}{4} + \frac{9 {\color{red}{v}}}{2} = \frac{9 \sin{\left(2 {\color{red}{\operatorname{asin}{\left(\frac{u}{3} \right)}}} \right)}}{4} + \frac{9 {\color{red}{\operatorname{asin}{\left(\frac{u}{3} \right)}}}}{2}$$

Kom ihåg att $$$u=x - 3$$$:

$$\frac{9 \sin{\left(2 \operatorname{asin}{\left(\frac{{\color{red}{u}}}{3} \right)} \right)}}{4} + \frac{9 \operatorname{asin}{\left(\frac{{\color{red}{u}}}{3} \right)}}{2} = \frac{9 \sin{\left(2 \operatorname{asin}{\left(\frac{{\color{red}{\left(x - 3\right)}}}{3} \right)} \right)}}{4} + \frac{9 \operatorname{asin}{\left(\frac{{\color{red}{\left(x - 3\right)}}}{3} \right)}}{2}$$

Alltså,

$$\int{\sqrt{- x^{2} + 6 x} d x} = \frac{9 \sin{\left(2 \operatorname{asin}{\left(\frac{x}{3} - 1 \right)} \right)}}{4} + \frac{9 \operatorname{asin}{\left(\frac{x}{3} - 1 \right)}}{2}$$

Använd formlerna $$$\sin{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\sin{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\cos{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 1 - 2 \alpha^{2}$$$, $$$\cos{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$, $$$\sinh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha^{2} + 1}$$$, $$$\sinh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha - 1} \sqrt{\alpha + 1}$$$, $$$\cosh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} + 1$$$, $$$\cosh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$ för att förenkla uttrycket:

$$\int{\sqrt{- x^{2} + 6 x} d x} = \frac{9 \sqrt{1 - \left(\frac{x}{3} - 1\right)^{2}} \left(\frac{x}{3} - 1\right)}{2} + \frac{9 \operatorname{asin}{\left(\frac{x}{3} - 1 \right)}}{2}$$

Förenkla ytterligare:

$$\int{\sqrt{- x^{2} + 6 x} d x} = \frac{\sqrt{9 - \left(x - 3\right)^{2}} \left(x - 3\right)}{2} + \frac{9 \operatorname{asin}{\left(\frac{x}{3} - 1 \right)}}{2}$$

Lägg till integrationskonstanten:

$$\int{\sqrt{- x^{2} + 6 x} d x} = \frac{\sqrt{9 - \left(x - 3\right)^{2}} \left(x - 3\right)}{2} + \frac{9 \operatorname{asin}{\left(\frac{x}{3} - 1 \right)}}{2}+C$$

$$$\int \sqrt{- x^{2} + 6 x}\, dx = \left(\frac{\sqrt{9 - \left(x - 3\right)^{2}} \left(x - 3\right)}{2} + \frac{9 \operatorname{asin}{\left(\frac{x}{3} - 1 \right)}}{2}\right) + C$$$A