Integral de $$$\sqrt{a^{2} - u^{2}}$$$ em relação a $$$u$$$

Encontre $$$\int \sqrt{a^{2} - u^{2}}\, du$$$.

Seja $$$u=\sin{\left(v \right)} \left|{a}\right|$$$.

Então $$$du=\left(\sin{\left(v \right)} \left|{a}\right|\right)^{\prime }dv = \cos{\left(v \right)} \left|{a}\right| dv$$$ (os passos podem ser vistos »).

Além disso, segue-se que $$$v=\operatorname{asin}{\left(\frac{u}{\left|{a}\right|} \right)}$$$.

Portanto,

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

Use a identidade $$$1 - \sin^{2}{\left( v \right)} = \cos^{2}{\left( v \right)}$$$:

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

Supondo que $$$\cos{\left( v \right)} \ge 0$$$, obtemos o seguinte:

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

Logo,

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

Aplique a fórmula de redução de potência $$$\cos^{2}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{1}{2}$$$ com $$$\alpha= v $$$:

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

Aplique a regra do múltiplo constante $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ usando $$$c=\frac{1}{2}$$$ e $$$f{\left(v \right)} = a^{2} \left(\cos{\left(2 v \right)} + 1\right)$$$:

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

Expand the expression:

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

Integre termo a termo:

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

Aplique a regra da constante $$$\int c\, dv = c v$$$ usando $$$c=a^{2}$$$:

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

Aplique a regra do múltiplo constante $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ usando $$$c=a^{2}$$$ e $$$f{\left(v \right)} = \cos{\left(2 v \right)}$$$:

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

Seja $$$w=2 v$$$.

Então $$$dw=\left(2 v\right)^{\prime }dv = 2 dv$$$ (veja os passos »), e obtemos $$$dv = \frac{dw}{2}$$$.

A integral pode ser reescrita como

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

Aplique a regra do múltiplo constante $$$\int c f{\left(w \right)}\, dw = c \int f{\left(w \right)}\, dw$$$ usando $$$c=\frac{1}{2}$$$ e $$$f{\left(w \right)} = \cos{\left(w \right)}$$$:

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

A integral do cosseno é $$$\int{\cos{\left(w \right)} d w} = \sin{\left(w \right)}$$$:

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

Recorde que $$$w=2 v$$$:

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

Recorde que $$$v=\operatorname{asin}{\left(\frac{u}{\left|{a}\right|} \right)}$$$:

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

Portanto,

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

Usando as fórmulas $$$\sin{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\sin{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\cos{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 1 - 2 \alpha^{2}$$$, $$$\cos{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$, $$$\sinh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha^{2} + 1}$$$, $$$\sinh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha - 1} \sqrt{\alpha + 1}$$$, $$$\cosh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} + 1$$$, $$$\cosh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$, simplifique a expressão:

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

Simplifique ainda mais:

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

Adicione a constante de integração:

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

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