We can solve the integral $\int\frac{1}{x\sqrt{1-x^2}}dx$ by applying integration method of trigonometric substitution using the substitution

Applying the trigonometric identity: $1-\sin\left(\theta \right)^2 = \cos\left(\theta \right)^2$

Simplify $\sqrt{\cos\left(\theta \right)^2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $\frac{1}{2}$

Simplify the fraction $\frac{\cos\left(\theta \right)}{\sin\left(\theta \right)\cos\left(\theta \right)}$ by $\cos\left(\theta \right)$

Applying the cosecant identity: $\displaystyle\csc\left(\theta\right)=\frac{1}{\sin\left(\theta\right)}$

The integral of $\csc(x)$ is $-\ln(\csc(x)+\cot(x))$

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$