Question

How do you evaluate `\int \frac { 1} { x \ln x) d x`?

Answer 1

`int \ 1/(xlnx) \ dx = ln|lnx| + C`

Explanation:

We seek:

`I = int \ 1/(xlnx) \ dx`

We can perform a substitution:

Let `u=lnx => (du)/dx = 1/x`

If we substitute this into the integral, we get:

`I = int \ 1/u \ du`

Which is a standard integral, so we integrate to get:

`I = ln|u| + C`

And we can restore the substitution, to get:

`I = ln|lnx| + C`