`int 1/sqrt(x^2-4) dx` Find the indefinite integral

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Let's apply integral substitution:`x=2sec(u)`





Now use the trigonometric identity: `tan^2(x)=sec^2(x)-1`



Now use the standard integral:`intsec(x)dx=ln|sec(x)+tan(x)|`

`=ln|sec(u)+tan(u)|` ----------(1)

Now from the substitution:

`sec(u)=x/2`  and,





Substitute back the above in the result  (1)




Since `ln(2)`  is constant, so we can omit it and add a new constant C to the solution,



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