Question

Intergal ((dx)/(x(sqrtlnx))

Answer 1

That looks fun... \[\large \int\limits \frac{ 1 }{x \sqrt{\ln x} } dx\]
But since the derivative of lnx is 1/x, I'd try u-substitution:

let \[\large u = \ln x\]
\[\large du = \frac{ 1 }{ x } dx\]

Answer 2

Just rearrange it a little so you can see it: \[\large \int\limits\limits \frac{ 1 }{x \sqrt{\ln x} } dx = \int\limits \frac{ 1 }{\sqrt {\ln x} } \times \frac{ 1 }{ x } dx \] \[\large = \int\limits \frac { 1 }{ \sqrt u } \frac{ 1 }{ x } dx = \int\limits \frac{ 1 }{\sqrt u } du\] \[\LARGE = \int\limits u^{-\frac{ 1 }{ 2 }} du\]
Boom. Now just increase the power by one and divide by the new power, add C, and replace u with ln x.