Question

How do you evaluate the integral `int 1/(x(lnx)^2) dx` from 0 to 1 if it converges?

Answer 1

The definite integral is not convergent.

Explanation:

First solve the indefinite integral using the substitution:

`x=e^t`
`dx=e^tdt`

`int dx/(x(lnx)^2) = int (e^tdt)/(e^t t^2) = int(dt)/t^2 = -1/(3t^3)=-1/(3(ln x)^3)`

The primitive is defined for `x in (0,1)` but:

`lim_(x->0^+) -1/(3(ln x)^3) = 0`

`lim_(x->1^-) -1/(3(ln x)^3) = +oo`

So the definite integral does not converge.