The problem is asking to find the definite integral of ln(9x)/x from 1 to e. To solve this quickly, we can use the formula for the integral of ln(ax) which is x(ln(ax) - 1).
However, as the limits of integration are from 1 to e and there's a 1/x outside of the ln(9x), this problem becomes a straightforward application of the first fundamental theorem of calculus.
Here are the steps for calculating this integral:
Step 1: Identify the integral that needs to be evaluated.
This integral is ∫ from 1 to e ln(9x)/x dx.
Step 2: Rewrite the integral in a more convenient form.
We rewrite the given integral into a more convenient form by factoring out the constant from the natural logarithm as: ∫ from 1 to e ln(9) + ln(x) dx.
Step 3: Break down the integral into simpler parts.
This is accomplished by splitting the integral into two terms so it can be easily solved: ∫ from 1 to e ln(9) dx + ∫ from 1 to e ln(x) dx.
Step 4: Evaluate each integral separately.
The integral of ln(9) with respect to x from 1 to e is e*ln(9) - ln(9) = ln(9)*(e - 1).
The integral of ln(x) with respect to x from 1 to e is e*ln(e) - 1*ln(1) = e - 0 = e.
Step 5: Combine the results.
We now add the values computed in the previous step: ln(9)*(e - 1) + e.
Upon performing the above steps, we determine that the integral ∫ from 1 to e ln(9x)/x dx is equal to 5.11112249896138 - log(9)^2/2.