How do you solve the integration of (sqrt(x/(x-1))-1)/x dx Do you know? I tried substitution, but with out any result... :/

Hello!

As I understand, we are speaking about

`int(sqrt(x/(x-1))-1)/x dx.`

We can omit the last term, -1/x, because we know its integral (ln|x|+C).

The remaining part is

`int sqrt(x/(x-1))/x dx=int 1/sqrt(x(x-1)) dx.`

Make some transformations with the function under integral, and assume that `xgt1:`

`1/sqrt(x(x-1))=1/sqrt(x(x-1))*(sqrt(x)+sqrt(x-1))/(sqrt(x)+sqrt(x-1))=`

`=(1/sqrt(x-1)+1/sqrt(x))/(sqrt(x)+sqrt(x-1))=`

`=2*((sqrt(x)+sqrt(x-1))')/(sqrt(x)+sqrt(x-1)).`

Now we can integrate it, because `int 1/u du=ln|u|+C:`

`int 1/sqrt(x(x-1)) dx = 2*ln(sqrt(x)+sqrt(x-1))+C.`

This is the answer for x>0 and without omitted `-ln(x).`