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

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

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

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

So,

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

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 »).

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:

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

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