30 Points! I need this fast! Thank you

Find the average value of the function g(x) = lnx/x over the interval [1, e].

Step-by-step explanation:

I don't often see calculus problems here!  I like it when I do!

The formula for the average value of a function is

Since e is greater than 1, our lower bound is 1 and the upper is e, and filling in our function we have this integral:

We'll use a u substitution to simplify before we integrate.  I teach my calculus students to rewrite the integral identifying what the dx is and therefore, what our du has to "match".  For example, the rewrite is:

Let u = ln(x), then

and

What our du is equal to is the same as what we have designated as our dx in the integral.  They "match". So I know I chose the right u.  In terms of u, our integral is

which integrates to

Making the back substitution of ln(x) for u gives us:

Using the First Fundamental Theorem of Calculus:

which simplifies down to

which finally becomes

Round to however many places you need.

Good luck in your calculus class!