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

Question

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?

Answer 1

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