Average Values

a. Verify that ∫ln x dx = x ln x - x + C.

b. Find the average value of ln x over [1, e].

Step 1:

In this question, we have to verify the given indefinite integral and find the average value of over .

Step 2:

(a) The given indefinite integral is .

. Now integrating by parts taking as the first function,

The integral is verified.