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?

Let
$$ g(y)=\int_{y}^{y+10}\log^2 x\,dx.\tag{1}$$
We have to compare $g(2),g(5)$ and $g(8)$. By the fundamental theorem of Calculus:
$$ g'(y) = \log^2(y+10)-\log^2(y) > 0, \tag{2}$$
hence $g$ is an increasing function and $g(2)<g(5)<g(8)$ (i.e. option $\color{red}{D}$) follows.
