Integral de $$$\frac{1}{\left(x^{2} - 1\right)^{\frac{3}{2}}}$$$

Encontre $$$\int \frac{1}{\left(x^{2} - 1\right)^{\frac{3}{2}}}\, dx$$$.

Seja $$$x=\cosh{\left(u \right)}$$$.

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

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

Portanto,

$$$\frac{1}{\left(x^{2} - 1\right)^{\frac{3}{2}}} = \frac{1}{\left(\cosh^{2}{\left( u \right)} - 1\right)^{\frac{3}{2}}}$$$

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

$$$\frac{1}{\left(\cosh^{2}{\left( u \right)} - 1\right)^{\frac{3}{2}}}=\frac{1}{\left(\sinh^{2}{\left( u \right)}\right)^{\frac{3}{2}}}$$$

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

$$$\frac{1}{\left(\sinh^{2}{\left( u \right)}\right)^{\frac{3}{2}}} = \frac{1}{\sinh^{3}{\left( u \right)}}$$$

Assim,

$${\color{red}{\int{\frac{1}{\left(x^{2} - 1\right)^{\frac{3}{2}}} d x}}} = {\color{red}{\int{\frac{1}{\sinh^{2}{\left(u \right)}} d u}}}$$

Reescreva o integrando em termos da cossecante hiperbólica:

$${\color{red}{\int{\frac{1}{\sinh^{2}{\left(u \right)}} d u}}} = {\color{red}{\int{\operatorname{csch}^{2}{\left(u \right)} d u}}}$$

A integral de $$$\operatorname{csch}^{2}{\left(u \right)}$$$ é $$$\int{\operatorname{csch}^{2}{\left(u \right)} d u} = - \coth{\left(u \right)}$$$:

$${\color{red}{\int{\operatorname{csch}^{2}{\left(u \right)} d u}}} = {\color{red}{\left(- \coth{\left(u \right)}\right)}}$$

Recorde que $$$u=\operatorname{acosh}{\left(x \right)}$$$:

$$- \coth{\left({\color{red}{u}} \right)} = - \coth{\left({\color{red}{\operatorname{acosh}{\left(x \right)}}} \right)}$$

Portanto,

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

Adicione a constante de integração:

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

$$$\int \frac{1}{\left(x^{2} - 1\right)^{\frac{3}{2}}}\, dx = - \frac{x}{\sqrt{x - 1} \sqrt{x + 1}} + C$$$A