Integral de $$$\sqrt{x^{2} - 6}$$$

Halla $$$\int \sqrt{x^{2} - 6}\, dx$$$.

Sea $$$x=\sqrt{6} \cosh{\left(u \right)}$$$.

Entonces $$$dx=\left(\sqrt{6} \cosh{\left(u \right)}\right)^{\prime }du = \sqrt{6} \sinh{\left(u \right)} du$$$ (los pasos pueden verse »).

Además, se sigue que $$$u=\operatorname{acosh}{\left(\frac{\sqrt{6} x}{6} \right)}$$$.

Por lo tanto,

$$$\sqrt{x^{2} - 6} = \sqrt{6 \cosh^{2}{\left( u \right)} - 6}$$$

Utiliza la identidad $$$\cosh^{2}{\left( u \right)} - 1 = \sinh^{2}{\left( u \right)}$$$:

$$$\sqrt{6 \cosh^{2}{\left( u \right)} - 6}=\sqrt{6} \sqrt{\cosh^{2}{\left( u \right)} - 1}=\sqrt{6} \sqrt{\sinh^{2}{\left( u \right)}}$$$

Suponiendo que $$$\sinh{\left( u \right)} \ge 0$$$, obtenemos lo siguiente:

$$$\sqrt{6} \sqrt{\sinh^{2}{\left( u \right)}} = \sqrt{6} \sinh{\left( u \right)}$$$

La integral se convierte en

$${\color{red}{\int{\sqrt{x^{2} - 6} d x}}} = {\color{red}{\int{6 \sinh^{2}{\left(u \right)} d u}}}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=6$$$ y $$$f{\left(u \right)} = \sinh^{2}{\left(u \right)}$$$:

$${\color{red}{\int{6 \sinh^{2}{\left(u \right)} d u}}} = {\color{red}{\left(6 \int{\sinh^{2}{\left(u \right)} d u}\right)}}$$

Aplica la fórmula de reducción de potencia $$$\sinh^{2}{\left(\alpha \right)} = \frac{\cosh{\left(2 \alpha \right)}}{2} - \frac{1}{2}$$$ con $$$\alpha= u $$$:

$$6 {\color{red}{\int{\sinh^{2}{\left(u \right)} d u}}} = 6 {\color{red}{\int{\left(\frac{\cosh{\left(2 u \right)}}{2} - \frac{1}{2}\right)d u}}}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{2}$$$ y $$$f{\left(u \right)} = \cosh{\left(2 u \right)} - 1$$$:

$$6 {\color{red}{\int{\left(\frac{\cosh{\left(2 u \right)}}{2} - \frac{1}{2}\right)d u}}} = 6 {\color{red}{\left(\frac{\int{\left(\cosh{\left(2 u \right)} - 1\right)d u}}{2}\right)}}$$

Integra término a término:

$$3 {\color{red}{\int{\left(\cosh{\left(2 u \right)} - 1\right)d u}}} = 3 {\color{red}{\left(- \int{1 d u} + \int{\cosh{\left(2 u \right)} d u}\right)}}$$

Aplica la regla de la constante $$$\int c\, du = c u$$$ con $$$c=1$$$:

$$3 \int{\cosh{\left(2 u \right)} d u} - 3 {\color{red}{\int{1 d u}}} = 3 \int{\cosh{\left(2 u \right)} d u} - 3 {\color{red}{u}}$$

Sea $$$v=2 u$$$.

Entonces $$$dv=\left(2 u\right)^{\prime }du = 2 du$$$ (los pasos pueden verse »), y obtenemos que $$$du = \frac{dv}{2}$$$.

Entonces,

$$- 3 u + 3 {\color{red}{\int{\cosh{\left(2 u \right)} d u}}} = - 3 u + 3 {\color{red}{\int{\frac{\cosh{\left(v \right)}}{2} d v}}}$$

Aplica la regla del factor constante $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ con $$$c=\frac{1}{2}$$$ y $$$f{\left(v \right)} = \cosh{\left(v \right)}$$$:

$$- 3 u + 3 {\color{red}{\int{\frac{\cosh{\left(v \right)}}{2} d v}}} = - 3 u + 3 {\color{red}{\left(\frac{\int{\cosh{\left(v \right)} d v}}{2}\right)}}$$

La integral del coseno hiperbólico es $$$\int{\cosh{\left(v \right)} d v} = \sinh{\left(v \right)}$$$:

$$- 3 u + \frac{3 {\color{red}{\int{\cosh{\left(v \right)} d v}}}}{2} = - 3 u + \frac{3 {\color{red}{\sinh{\left(v \right)}}}}{2}$$

Recordemos que $$$v=2 u$$$:

$$- 3 u + \frac{3 \sinh{\left({\color{red}{v}} \right)}}{2} = - 3 u + \frac{3 \sinh{\left({\color{red}{\left(2 u\right)}} \right)}}{2}$$

Recordemos que $$$u=\operatorname{acosh}{\left(\frac{\sqrt{6} x}{6} \right)}$$$:

$$\frac{3 \sinh{\left(2 {\color{red}{u}} \right)}}{2} - 3 {\color{red}{u}} = \frac{3 \sinh{\left(2 {\color{red}{\operatorname{acosh}{\left(\frac{\sqrt{6} x}{6} \right)}}} \right)}}{2} - 3 {\color{red}{\operatorname{acosh}{\left(\frac{\sqrt{6} x}{6} \right)}}}$$

Por lo tanto,

$$\int{\sqrt{x^{2} - 6} d x} = \frac{3 \sinh{\left(2 \operatorname{acosh}{\left(\frac{\sqrt{6} x}{6} \right)} \right)}}{2} - 3 \operatorname{acosh}{\left(\frac{\sqrt{6} x}{6} \right)}$$

Usando las fórmulas $$$\sin{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\sin{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\cos{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 1 - 2 \alpha^{2}$$$, $$$\cos{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$, $$$\sinh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha^{2} + 1}$$$, $$$\sinh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha - 1} \sqrt{\alpha + 1}$$$, $$$\cosh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} + 1$$$, $$$\cosh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$, simplifica la expresión:

$$\int{\sqrt{x^{2} - 6} d x} = \frac{\sqrt{6} x \sqrt{\frac{\sqrt{6} x}{6} - 1} \sqrt{\frac{\sqrt{6} x}{6} + 1}}{2} - 3 \operatorname{acosh}{\left(\frac{\sqrt{6} x}{6} \right)}$$

Simplificar más:

$$\int{\sqrt{x^{2} - 6} d x} = \frac{\sqrt{6} x \sqrt{\sqrt{6} x - 6} \sqrt{\sqrt{6} x + 6}}{12} - 3 \operatorname{acosh}{\left(\frac{\sqrt{6} x}{6} \right)}$$

Añade la constante de integración:

$$\int{\sqrt{x^{2} - 6} d x} = \frac{\sqrt{6} x \sqrt{\sqrt{6} x - 6} \sqrt{\sqrt{6} x + 6}}{12} - 3 \operatorname{acosh}{\left(\frac{\sqrt{6} x}{6} \right)}+C$$

$$$\int \sqrt{x^{2} - 6}\, dx = \left(\frac{\sqrt{6} x \sqrt{\sqrt{6} x - 6} \sqrt{\sqrt{6} x + 6}}{12} - 3 \operatorname{acosh}{\left(\frac{\sqrt{6} x}{6} \right)}\right) + C$$$A