Let $ f(x)= ( \log_e x) ^2 $ and (Integration by parts. Comparing integrals of different limits ) Let $ f(x)=( \log_e x) ^2 $ and let 
$ I_1= \int_{2}^{12} f(x) dx $  ,  $ I_2= \int_{5}^{15} f(x) dx $ and $ \int_{8}^{18} f(x) dx$
Then which of the following is true?
(A)$I_3 <I_1 < I_2 $
(B)$I_2 <I_3 < I_1 $
(C)$I_3 <I_2 < I_1 $
(D)$I_1 <I_2 < I_3 $
What I've done: 
Have computed the integration (by integration by parts) and computed $ I_1, I_2 $ and $ I_3 $ with the help of a calculator. So option D is correct. Also intuitively $f(x)$ is increasing function. Option D is obvious. 
But I want to know how to do without calculating the limits? Calculator won't be allowed in the test. What's the intuition?

Note that each of the integrals are over an interval of width $10$, and so to see which integral has a larger value, we want to find the interval in which $f$ has the largest average value. Because $f$ is increasing and $2<5<8$, we must have $I_1<I_2<I_3$.
