JEE MAIN - Mathematics (2005 - No. 20)
If $${I_1} = \int\limits_0^1 {{2^{{x^2}}}dx,{I_2} = \int\limits_0^1 {{2^{{x^3}}}dx,\,{I_3} = \int\limits_1^2 {{2^{{x^2}}}dx} } } $$ and $${I_4} = \int\limits_1^2 {{2^{{x^3}}}dx} $$ then
$${I_2} > {I_1}$$
$${I_1} > {I_2}$$
$${I_3} = {I_4}$$
$${I_3} > {I_4}$$
Explanation
$${I_1} = \int\limits_0^1 {{2^{{x^2}}}} dx,\,{I_2} = \int\limits_0^1 {{2^{{x^3}}}} dx,$$
$$ = {I_3} = \int\limits_0^1 {{2^{{x^2}}}} dx,\,$$
$${I_4} = \int\limits_0^1 {{2^{{x^3}}}} dx\,\,$$
$$\forall 0 < x < 1,\,{x^2} > {x^3}$$
$$ \Rightarrow \int\limits_0^1 {{2^{{x^2}}}} \,dx > \int\limits_0^1 {{2^{{x^3}}}} dx$$
and $$\int\limits_1^2 {{2^{{x^3}}}dx} > \int\limits_1^2 {{2^{{x^2}}}dx} $$
$$ \Rightarrow {I_1} > {I_2}$$ and $${I_4} > {I_3}$$
$$ = {I_3} = \int\limits_0^1 {{2^{{x^2}}}} dx,\,$$
$${I_4} = \int\limits_0^1 {{2^{{x^3}}}} dx\,\,$$
$$\forall 0 < x < 1,\,{x^2} > {x^3}$$
$$ \Rightarrow \int\limits_0^1 {{2^{{x^2}}}} \,dx > \int\limits_0^1 {{2^{{x^3}}}} dx$$
and $$\int\limits_1^2 {{2^{{x^3}}}dx} > \int\limits_1^2 {{2^{{x^2}}}dx} $$
$$ \Rightarrow {I_1} > {I_2}$$ and $${I_4} > {I_3}$$
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