Integrating $\sqrt{\frac{1}{a}-\frac{1}{x}}$ .

Question

How do I integrate the following algebraic function?

$\sqrt{\frac{1}{a} - \frac{1}{x}}$

Answer 1

Integrate by parts to get

$$x\sqrt{\frac{1}{a}-\frac{1}{x}} - \frac{1}{2}\int \left(\frac{1}{a}-\frac{1}{x}\right)^{-1/2}\frac{1}{x} \; dx.$$

The last integral can me made into

$$\int \frac{\sqrt{a}}{\sqrt{x^2-ax}} \; dx,$$

so complete the square and substitute $u=x-a/2$ to get

$$\int \frac{\sqrt{a}}{\sqrt{u^2-a^2/4}} \; du$$

which, by standard techniques becomes

$$\sqrt{a}\ln ( 2 u +\sqrt{4u^2-a^2}).$$

Then plug everything in. And may God have mercy on your soul.

Answer 2

substituting $$t=\sqrt{\frac{1}{a}-\frac{1}{x}}$$ then we get $$x=\frac{a}{1-at^2}$$ and $$dx=\frac{2at}{(1-at^2)^{2}}dt$$ this is a better way!and so we get $$\int\frac{2a^2t^2}{(1-at^2)^2}dt$$ Can you solve this? we have $$x=a(1-at^2)^{-1}$$ then $$dx=a(-1)(1-at^2)^{-2}(-2at)dt$$ is it clear now?