integral of 1 to e^4 dx/x(1+lnx)

Question

1+lnx = a => dx/x = da
dx/ x(1+lnx) = da/a
int of da/a = In(a)

=>int of dx/ x(1+lnx) = In(1+In(x)) + C

from 1 to e^4 for int of dx/ x(1+lnx) = In(1+In(1)) - In(1+In(e^4))
In(1+In(1)) - In(1+In(e^4)) = -In(5) =~ -1.60944

I am getting this as the wrong answer am i doing something wrong?

Answer 1

It all looks OK to me, except from 1 to e^4 you should have your signs the other way around:

ln(1+ln(e^4)) - ln(1+In(1))

...

=~ +1.60944