$$$\sqrt{a^{2} - x^{2}}$$$ の $$$x$$$ に関する積分

$$$\int \sqrt{a^{2} - x^{2}}\, dx$$$ を求めよ。

$$$x=\sin{\left(u \right)} \left|{a}\right|$$$ とする。

すると $$$dx=\left(\sin{\left(u \right)} \left|{a}\right|\right)^{\prime }du = \cos{\left(u \right)} \left|{a}\right| du$$$ (手順は»で確認できます)。

また、$$$u=\operatorname{asin}{\left(\frac{x}{\left|{a}\right|} \right)}$$$が成り立つ。

したがって、

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

恒等式 $$$1 - \sin^{2}{\left( u \right)} = \cos^{2}{\left( u \right)}$$$ を用いよ:

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

$$$\cos{\left( u \right)} \ge 0$$$ を仮定すると、以下が得られる:

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

したがって、

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

冪低減公式 $$$\cos^{2}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{1}{2}$$$ を $$$\alpha= u $$$ に適用する:

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

定数倍の法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ を、$$$c=\frac{1}{2}$$$ と $$$f{\left(u \right)} = a^{2} \left(\cos{\left(2 u \right)} + 1\right)$$$ に対して適用する:

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

Expand the expression:

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

項別に積分せよ:

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

$$$c=a^{2}$$$ に対して定数則 $$$\int c\, du = c u$$$ を適用する:

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

定数倍の法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ を、$$$c=a^{2}$$$ と $$$f{\left(u \right)} = \cos{\left(2 u \right)}$$$ に対して適用する:

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

$$$v=2 u$$$ とする。

すると $$$dv=\left(2 u\right)^{\prime }du = 2 du$$$（手順は»で確認できます）、$$$du = \frac{dv}{2}$$$ となります。

したがって、

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

定数倍の法則 $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ を、$$$c=\frac{1}{2}$$$ と $$$f{\left(v \right)} = \cos{\left(v \right)}$$$ に対して適用する:

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

余弦の積分は$$$\int{\cos{\left(v \right)} d v} = \sin{\left(v \right)}$$$:

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

次のことを思い出してください $$$v=2 u$$$:

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

次のことを思い出してください $$$u=\operatorname{asin}{\left(\frac{x}{\left|{a}\right|} \right)}$$$:

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

したがって、

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

公式 $$$\sin{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\sin{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\cos{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 1 - 2 \alpha^{2}$$$, $$$\cos{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$, $$$\sinh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha^{2} + 1}$$$, $$$\sinh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha - 1} \sqrt{\alpha + 1}$$$, $$$\cosh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} + 1$$$, $$$\cosh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$ を用いて、式を簡単化しなさい:

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

さらに簡単化:

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

積分定数を加える:

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

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