Integral of $$$\frac{1}{x \sqrt{x^{2} - 4}}$$$

Find $$$\int \frac{1}{x \sqrt{x^{2} - 4}}\, dx$$$.

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

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

The integral can be rewritten as

$${\color{red}{\int{\frac{1}{x \sqrt{x^{2} - 4}} d x}}} = {\color{red}{\int{\left(- \frac{1}{\sqrt{1 - 4 u^{2}}}\right)d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=-1$$$ and $$$f{\left(u \right)} = \frac{1}{\sqrt{1 - 4 u^{2}}}$$$:

$${\color{red}{\int{\left(- \frac{1}{\sqrt{1 - 4 u^{2}}}\right)d u}}} = {\color{red}{\left(- \int{\frac{1}{\sqrt{1 - 4 u^{2}}} d u}\right)}}$$

Let $$$u=\frac{\sin{\left(v \right)}}{2}$$$.

Then $$$du=\left(\frac{\sin{\left(v \right)}}{2}\right)^{\prime }dv = \frac{\cos{\left(v \right)}}{2} dv$$$ (steps can be seen »).

Also, it follows that $$$v=\operatorname{asin}{\left(2 u \right)}$$$.

Integrand becomes

$$$\frac{1}{\sqrt{1 - 4 u ^{2}}} = \frac{1}{\sqrt{1 - \sin^{2}{\left( v \right)}}}$$$

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:

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

So,

$$- {\color{red}{\int{\frac{1}{\sqrt{1 - 4 u^{2}}} d u}}} = - {\color{red}{\int{\frac{1}{2} d v}}}$$

Apply the constant rule $$$\int c\, dv = c v$$$ with $$$c=\frac{1}{2}$$$:

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

Recall that $$$v=\operatorname{asin}{\left(2 u \right)}$$$:

$$- \frac{{\color{red}{v}}}{2} = - \frac{{\color{red}{\operatorname{asin}{\left(2 u \right)}}}}{2}$$

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

$$- \frac{\operatorname{asin}{\left(2 {\color{red}{u}} \right)}}{2} = - \frac{\operatorname{asin}{\left(2 {\color{red}{\frac{1}{x}}} \right)}}{2}$$

Therefore,

$$\int{\frac{1}{x \sqrt{x^{2} - 4}} d x} = - \frac{\operatorname{asin}{\left(\frac{2}{x} \right)}}{2}$$

Add the constant of integration:

$$\int{\frac{1}{x \sqrt{x^{2} - 4}} d x} = - \frac{\operatorname{asin}{\left(\frac{2}{x} \right)}}{2}+C$$

$$$\int \frac{1}{x \sqrt{x^{2} - 4}}\, dx = - \frac{\operatorname{asin}{\left(\frac{2}{x} \right)}}{2} + C$$$A