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I'm having trouble with the following integral:

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

I attempted the substitution $u = a-x^{-1}$ to get

$$\int \frac{1}{\sqrt{u} \, (a-u)^2} \, du,$$

but I don't see where to go from there. Can someone give me a hint in the right direction?

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If I had seen the original integral and not seen your attempted substitution, my first inclination would have been to just set $u^2=a-x^{-1}$, solve for $x$ to get $dx$, and proceed from there. This brings you to the same place as DonAntonio's answer, and is a good strategy for situations like that. Think of the objective - to get rid of the square root operation. – process91 Jun 4 '12 at 3:35

You're almost there. Now you can do$$u=t^2\Longrightarrow du=2tdt$$ and the last integral becomes $$\int\frac{2\rlap{/}{t}\,dt}{\rlap{/}{t}(a-t^2)^2}$$ which is the integral of a rational function

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