Evaluate the definite integral? Integral (e^8 upper bound, 1 lower bound) for [dx/(x(1+lnx))]

First, we have the expression:

dx / (x(1 + lnx)

In order to integrate this, we will use u-substitution. Let u = (1 + lnx), and so:
du = dx/x

Thus, we can re-write this as:

du/u

This can be easily integrated to be:

ln(u)

Now, plugging back in our value from before we substituted, the solution is:

ln(1 + lnx)

Now, we simply plug in the upper and lower bounds:

(ln(1 + ln(e^8))) - (ln(1 + ln(1)))

ln(1 + 8) - ln(1 + 0)

ln(9) - ln(1)

ln(9) - 0

The solution is ln(9) or about 2.197