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

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

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

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

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

Alltså,

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

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

$$$\sqrt{a^{2} \cosh^{2}{\left( u \right)} - a^{2}}=\sqrt{\cosh^{2}{\left( u \right)} - 1} \left|{a}\right|=\sqrt{\sinh^{2}{\left( u \right)}} \left|{a}\right|$$$

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

$$$\sqrt{\sinh^{2}{\left( u \right)}} \left|{a}\right| = \sinh{\left( u \right)} \left|{a}\right|$$$

Integralen kan skrivas om som

$${\color{red}{\int{\sqrt{- a^{2} + x^{2}} d x}}} = {\color{red}{\int{a^{2} \sinh^{2}{\left(u \right)} d u}}}$$

Använd potensreduceringsformeln $$$\sinh^{2}{\left(\alpha \right)} = \frac{\cosh{\left(2 \alpha \right)}}{2} - \frac{1}{2}$$$ med $$$\alpha= u $$$:

$${\color{red}{\int{a^{2} \sinh^{2}{\left(u \right)} d u}}} = {\color{red}{\int{\frac{a^{2} \left(\cosh{\left(2 u \right)} - 1\right)}{2} d u}}}$$

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

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

Expand the expression:

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

Integrera termvis:

$$\frac{{\color{red}{\int{\left(a^{2} \cosh{\left(2 u \right)} - a^{2}\right)d u}}}}{2} = \frac{{\color{red}{\left(- \int{a^{2} d u} + \int{a^{2} \cosh{\left(2 u \right)} d u}\right)}}}{2}$$

Tillämpa konstantregeln $$$\int c\, du = c u$$$ med $$$c=a^{2}$$$:

$$\frac{\int{a^{2} \cosh{\left(2 u \right)} d u}}{2} - \frac{{\color{red}{\int{a^{2} d u}}}}{2} = \frac{\int{a^{2} \cosh{\left(2 u \right)} d u}}{2} - \frac{{\color{red}{a^{2} u}}}{2}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=a^{2}$$$ och $$$f{\left(u \right)} = \cosh{\left(2 u \right)}$$$:

$$- \frac{a^{2} u}{2} + \frac{{\color{red}{\int{a^{2} \cosh{\left(2 u \right)} d u}}}}{2} = - \frac{a^{2} u}{2} + \frac{{\color{red}{a^{2} \int{\cosh{\left(2 u \right)} d u}}}}{2}$$

Låt $$$v=2 u$$$ vara.

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

Alltså,

$$- \frac{a^{2} u}{2} + \frac{a^{2} {\color{red}{\int{\cosh{\left(2 u \right)} d u}}}}{2} = - \frac{a^{2} u}{2} + \frac{a^{2} {\color{red}{\int{\frac{\cosh{\left(v \right)}}{2} d v}}}}{2}$$

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

$$- \frac{a^{2} u}{2} + \frac{a^{2} {\color{red}{\int{\frac{\cosh{\left(v \right)}}{2} d v}}}}{2} = - \frac{a^{2} u}{2} + \frac{a^{2} {\color{red}{\left(\frac{\int{\cosh{\left(v \right)} d v}}{2}\right)}}}{2}$$

Integralen av den hyperboliska cosinusfunktionen är $$$\int{\cosh{\left(v \right)} d v} = \sinh{\left(v \right)}$$$:

$$- \frac{a^{2} u}{2} + \frac{a^{2} {\color{red}{\int{\cosh{\left(v \right)} d v}}}}{4} = - \frac{a^{2} u}{2} + \frac{a^{2} {\color{red}{\sinh{\left(v \right)}}}}{4}$$

Kom ihåg att $$$v=2 u$$$:

$$- \frac{a^{2} u}{2} + \frac{a^{2} \sinh{\left({\color{red}{v}} \right)}}{4} = - \frac{a^{2} u}{2} + \frac{a^{2} \sinh{\left({\color{red}{\left(2 u\right)}} \right)}}{4}$$

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

$$\frac{a^{2} \sinh{\left(2 {\color{red}{u}} \right)}}{4} - \frac{a^{2} {\color{red}{u}}}{2} = \frac{a^{2} \sinh{\left(2 {\color{red}{\operatorname{acosh}{\left(\frac{x}{\left|{a}\right|} \right)}}} \right)}}{4} - \frac{a^{2} {\color{red}{\operatorname{acosh}{\left(\frac{x}{\left|{a}\right|} \right)}}}}{2}$$

Alltså,

$$\int{\sqrt{- a^{2} + x^{2}} d x} = \frac{a^{2} \sinh{\left(2 \operatorname{acosh}{\left(\frac{x}{\left|{a}\right|} \right)} \right)}}{4} - \frac{a^{2} \operatorname{acosh}{\left(\frac{x}{\left|{a}\right|} \right)}}{2}$$

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:

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

Lägg till integrationskonstanten:

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

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