Integraal van $$$\sqrt{b^{2} - x^{2}}$$$ met betrekking tot $$$x$$$

Bepaal $$$\int \sqrt{b^{2} - x^{2}}\, dx$$$.

Zij $$$x=\sin{\left(u \right)} \left|{b}\right|$$$.

Dan $$$dx=\left(\sin{\left(u \right)} \left|{b}\right|\right)^{\prime }du = \cos{\left(u \right)} \left|{b}\right| du$$$ (zie » voor de stappen).

Bovendien volgt dat $$$u=\operatorname{asin}{\left(\frac{x}{\left|{b}\right|} \right)}$$$.

Dus,

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

Gebruik de identiteit $$$1 - \sin^{2}{\left( u \right)} = \cos^{2}{\left( u \right)}$$$:

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

Aangenomen dat $$$\cos{\left( u \right)} \ge 0$$$, verkrijgen we het volgende:

$$$\sqrt{\cos^{2}{\left( u \right)}} \left|{b}\right| = \cos{\left( u \right)} \left|{b}\right|$$$

Dus,

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

Pas de machtsreductieformule $$$\cos^{2}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{1}{2}$$$ toe met $$$\alpha= u $$$:

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

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=\frac{1}{2}$$$ en $$$f{\left(u \right)} = b^{2} \left(\cos{\left(2 u \right)} + 1\right)$$$:

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

Expand the expression:

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

Integreer termgewijs:

$$\frac{{\color{red}{\int{\left(b^{2} \cos{\left(2 u \right)} + b^{2}\right)d u}}}}{2} = \frac{{\color{red}{\left(\int{b^{2} d u} + \int{b^{2} \cos{\left(2 u \right)} d u}\right)}}}{2}$$

Pas de constantenregel $$$\int c\, du = c u$$$ toe met $$$c=b^{2}$$$:

$$\frac{\int{b^{2} \cos{\left(2 u \right)} d u}}{2} + \frac{{\color{red}{\int{b^{2} d u}}}}{2} = \frac{\int{b^{2} \cos{\left(2 u \right)} d u}}{2} + \frac{{\color{red}{b^{2} u}}}{2}$$

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=b^{2}$$$ en $$$f{\left(u \right)} = \cos{\left(2 u \right)}$$$:

$$\frac{b^{2} u}{2} + \frac{{\color{red}{\int{b^{2} \cos{\left(2 u \right)} d u}}}}{2} = \frac{b^{2} u}{2} + \frac{{\color{red}{b^{2} \int{\cos{\left(2 u \right)} d u}}}}{2}$$

Zij $$$v=2 u$$$.

Dan $$$dv=\left(2 u\right)^{\prime }du = 2 du$$$ (de stappen zijn te zien »), en dan geldt dat $$$du = \frac{dv}{2}$$$.

De integraal wordt

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

Pas de constante-veelvoudregel $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ toe met $$$c=\frac{1}{2}$$$ en $$$f{\left(v \right)} = \cos{\left(v \right)}$$$:

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

De integraal van de cosinus is $$$\int{\cos{\left(v \right)} d v} = \sin{\left(v \right)}$$$:

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

We herinneren eraan dat $$$v=2 u$$$:

$$\frac{b^{2} u}{2} + \frac{b^{2} \sin{\left({\color{red}{v}} \right)}}{4} = \frac{b^{2} u}{2} + \frac{b^{2} \sin{\left({\color{red}{\left(2 u\right)}} \right)}}{4}$$

We herinneren eraan dat $$$u=\operatorname{asin}{\left(\frac{x}{\left|{b}\right|} \right)}$$$:

$$\frac{b^{2} \sin{\left(2 {\color{red}{u}} \right)}}{4} + \frac{b^{2} {\color{red}{u}}}{2} = \frac{b^{2} \sin{\left(2 {\color{red}{\operatorname{asin}{\left(\frac{x}{\left|{b}\right|} \right)}}} \right)}}{4} + \frac{b^{2} {\color{red}{\operatorname{asin}{\left(\frac{x}{\left|{b}\right|} \right)}}}}{2}$$

Dus,

$$\int{\sqrt{b^{2} - x^{2}} d x} = \frac{b^{2} \sin{\left(2 \operatorname{asin}{\left(\frac{x}{\left|{b}\right|} \right)} \right)}}{4} + \frac{b^{2} \operatorname{asin}{\left(\frac{x}{\left|{b}\right|} \right)}}{2}$$

Gebruik de formules $$$\sin{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\sin{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\cos{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 1 - 2 \alpha^{2}$$$, $$$\cos{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$, $$$\sinh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha^{2} + 1}$$$, $$$\sinh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha - 1} \sqrt{\alpha + 1}$$$, $$$\cosh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} + 1$$$, $$$\cosh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$ om de uitdrukking te vereenvoudigen:

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

Vereenvoudig verder:

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

Voeg de integratieconstante toe:

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

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