Integral de $$$\sqrt{x^{2} + y^{2}}$$$ em relação a $$$x$$$

Encontre $$$\int \sqrt{x^{2} + y^{2}}\, dx$$$.

Seja $$$x=\sinh{\left(u \right)} \left|{y}\right|$$$.

Então $$$dx=\left(\sinh{\left(u \right)} \left|{y}\right|\right)^{\prime }du = \cosh{\left(u \right)} \left|{y}\right| du$$$ (os passos podem ser vistos »).

Além disso, segue-se que $$$u=\operatorname{asinh}{\left(\frac{x}{\left|{y}\right|} \right)}$$$.

Logo,

$$$\sqrt{x^{2} + y^{2}} = \sqrt{y^{2} \sinh^{2}{\left( u \right)} + y^{2}}$$$

Use a identidade $$$\sinh^{2}{\left( u \right)} + 1 = \cosh^{2}{\left( u \right)}$$$:

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

$$$\sqrt{\cosh^{2}{\left( u \right)}} \left|{y}\right| = \cosh{\left( u \right)} \left|{y}\right|$$$

Portanto,

$${\color{red}{\int{\sqrt{x^{2} + y^{2}} d x}}} = {\color{red}{\int{y^{2} \cosh^{2}{\left(u \right)} d u}}}$$

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

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

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{1}{2}$$$ e $$$f{\left(u \right)} = y^{2} \left(\cosh{\left(2 u \right)} + 1\right)$$$:

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

Expand the expression:

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

Integre termo a termo:

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

Aplique a regra da constante $$$\int c\, du = c u$$$ usando $$$c=y^{2}$$$:

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

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=y^{2}$$$ e $$$f{\left(u \right)} = \cosh{\left(2 u \right)}$$$:

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

Seja $$$v=2 u$$$.

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

A integral pode ser reescrita como

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

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 $$$f{\left(v \right)} = \cosh{\left(v \right)}$$$:

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

A integral do cosseno hiperbólico é $$$\int{\cosh{\left(v \right)} d v} = \sinh{\left(v \right)}$$$:

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

Recorde que $$$v=2 u$$$:

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

Recorde que $$$u=\operatorname{asinh}{\left(\frac{x}{\left|{y}\right|} \right)}$$$:

$$\frac{y^{2} \sinh{\left(2 {\color{red}{u}} \right)}}{4} + \frac{y^{2} {\color{red}{u}}}{2} = \frac{y^{2} \sinh{\left(2 {\color{red}{\operatorname{asinh}{\left(\frac{x}{\left|{y}\right|} \right)}}} \right)}}{4} + \frac{y^{2} {\color{red}{\operatorname{asinh}{\left(\frac{x}{\left|{y}\right|} \right)}}}}{2}$$

Portanto,

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

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:

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

Simplifique ainda mais:

$$\int{\sqrt{x^{2} + y^{2}} d x} = \frac{x \sqrt{x^{2} + y^{2}} + y^{2} \operatorname{asinh}{\left(\frac{x}{\left|{y}\right|} \right)}}{2}$$

Adicione a constante de integração:

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

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