Aspire Faculty ID #3632 · Topic: NIMCET 2012 · Just now
NIMCET 2012

If $ I_1 = \displaystyle \int_{0}^{1} 2^{x^2},dx,\quad I_2 = \displaystyle \int_{0}^{1} 2^{x^3},dx,\quad I_3 = \displaystyle \int_{1}^{2} 2^{x^2},dx,\quad I_4 = \displaystyle \int_{1}^{2} 2^{x^3},dx,$ then

Solution

On the interval $0 \le x \le 1$: 
$ x^3 < x^2 \Rightarrow 2^{x^3} < 2^{x^2} $ 
So, $ I_2 = \displaystyle \int_0^1 2^{x^3},dx < \int_0^1 2^{x^2}dx = I_1 $ 
Thus, $ I_1 > I_2 $ 

 On the interval $1 \le x \le 2$: 
$ x^3 > x^2 \Rightarrow 2^{x^3} > 2^{x^2} $ 
So, $ I_4 = \displaystyle \int_1^2 2^{x^3}dx > \int_1^2 2^{x^2}dx = I_3 $ 
Thus, $ I_4 > I_3 $
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