Evaluate the integral. (Use $$ C $$ for the constant of integration.) $$ \int \frac{\sqrt{x^{2}-4}}{x^{3}} d x $$ $$ \square $$

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Calculate the integral $$ \int \frac{\sqrt{x^{2}-4}}{x^{3}} d x $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int \frac{\sqrt{x^{2}-4}}{x^{3}} dx$$
- step1: Transform the expression:
  $$\int \frac{\sqrt{\sec^{2}\left(tight)-1}}{4\sec^{3}\left(tight)}	imes 2\sec\left(tight)	an\left(tight) dt$$
- step2: Simplify the expression:
  $$\int \frac{\sqrt{\sec^{2}\left(tight)-1}	imes 	an\left(tight)}{2\sec^{2}\left(tight)} dt$$
- step3: Simplify the expression:
  $$\int \frac{1}{4}\left|\sin\left(2tight)ight|	imes 	an\left(tight) dt$$
- step4: Remove the absolute value bars:
  $$\int \frac{1}{4}\sin\left(2tight)	an\left(tight) dt$$
- step5: Simplify the expression:
  $$\int \frac{1}{2}\sin^{2}\left(tight) dt$$
- step6: Use properties of integrals:
  $$\frac{1}{2}	imes \int \sin^{2}\left(tight) dt$$
- step7: Evaluate the integral:
  $$\frac{1}{2}\left(\frac{t}{2}-\frac{\sin\left(2tight)}{4}ight)$$
- step8: Simplify the expression:
  $$\frac{1}{2}\left(\frac{t}{2}-\frac{1}{4}\sin\left(2tight)ight)$$
- step9: Use the distributive property:
  $$\frac{1}{2}	imes \frac{t}{2}+\frac{1}{2}\left(-\frac{1}{4}\sin\left(2tight)ight)$$
- step10: Multiply the terms:
  $$\frac{t}{4}+\frac{1}{2}\left(-\frac{1}{4}\sin\left(2tight)ight)$$
- step11: Multiply the terms:
  $$\frac{t}{4}-\frac{1}{8}\sin\left(2tight)$$
- step12: Substitute back:
  $$\frac{\operatorname{arcsec}\left(\frac{1}{2}xight)}{4}-\frac{1}{8}\sin\left(2\operatorname{arcsec}\left(\frac{1}{2}xight)ight)$$
- step13: Simplify the expression:
  $$\frac{1}{4}\operatorname{arcsec}\left(\frac{1}{2}xight)-\frac{1}{8}\sin\left(2\operatorname{arcsec}\left(\frac{1}{2}xight)ight)$$
- step14: Add the constant of integral C:
  $$\frac{1}{4}\operatorname{arcsec}\left(\frac{1}{2}xight)-\frac{1}{8}\sin\left(2\operatorname{arcsec}\left(\frac{1}{2}xight)ight) + C, C \in \mathbb{R}$$
The integral of $$ \frac{\sqrt{x^{2}-4}}{x^{3}} $$ with respect to $$ x $$ is $$ \frac{1}{4}\operatorname{arcsec}(\frac{1}{2}x)-\frac{1}{8}\sin(2\operatorname{arcsec}(\frac{1}{2}x)) + C $$, where $$ C $$ is the constant of integration.
The integral $$ \int \frac{\sqrt{x^{2}-4}}{x^{3}} d x $$ equals $$ \frac{1}{4}\operatorname{arcsec}(\frac{1}{2}x)-\frac{1}{8}\sin(2\operatorname{arcsec}(\frac{1}{2}x)) + C $$, where $$ C $$ is the constant of integration.

Evaluate the integral. (Use for the constant of integration.)

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