Integral de $$$\frac{1}{x \sqrt{x^{2} - 1}}$$$

Encontre $$$\int \frac{1}{x \sqrt{x^{2} - 1}}\, dx$$$.

Let $$$u=\frac{1}{x}$$$.

Then $$$du=\left(\frac{1}{x}\right)^{\prime }dx = - \frac{1}{x^{2}} dx$$$ (steps can be seen here), and we have that $$$\frac{dx}{x^{2}} = - du$$$.

Therefore,

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

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=-1$$$ and $$$f{\left(u \right)} = \frac{1}{\sqrt{1 - u^{2}}}$$$:

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

Let $$$u=\sin{\left(v \right)}$$$.

Then $$$du=\left(\sin{\left(v \right)}\right)^{\prime }dv = \cos{\left(v \right)} dv$$$ (steps can be seen here).

Also, it follows that $$$v=\operatorname{asin}{\left(u \right)}$$$.

Therefore,

$$$\frac{1}{\sqrt{1 - u ^{2}}} = \frac{1}{\sqrt{1 - \sin^{2}{\left( v \right)}}}$$$

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

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

Assuming that $$$\cos{\left( v \right)} \ge 0$$$, we obtain the following:

$$$\frac{1}{\sqrt{\cos^{2}{\left( v \right)}}} = \frac{1}{\cos{\left( v \right)}}$$$

Integral becomes

$$- {\color{red}{\int{\frac{1}{\sqrt{1 - u^{2}}} d u}}} = - {\color{red}{\int{1 d v}}}$$

Apply the constant rule $$$\int c\, dv = c v$$$ with $$$c=1$$$:

$$- {\color{red}{\int{1 d v}}} = - {\color{red}{v}}$$

Recall that $$$v=\operatorname{asin}{\left(u \right)}$$$:

$$- {\color{red}{v}} = - {\color{red}{\operatorname{asin}{\left(u \right)}}}$$

Recall that $$$u=\frac{1}{x}$$$:

$$- \operatorname{asin}{\left({\color{red}{u}} \right)} = - \operatorname{asin}{\left({\color{red}{\frac{1}{x}}} \right)}$$

Therefore,

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

Add the constant of integration:

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

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