If I_1 = _0^1 2^x^2,dx, I_2 = _0^1 2^x^3,dx, I_3 = _1^2 2^x^2,dx and I_4 = _1^2 2^x^3,dx, …

Question

If $I_1 = \displaystyle\int_0^1 2^{x^2}\,dx$, $I_2 = \displaystyle\int_0^1 2^{x^3}\,dx$, $I_3 = \displaystyle\int_1^2 2^{x^2}\,dx$ and $I_4 = \displaystyle\int_1^2 2^{x^3}\,dx$, then:

Accepted Answer

$${I_1} = \int\limits_0^1 {{2^{{x^2}}}} dx,\,{I_2} = \int\limits_0^1 {{2^{{x^3}}}} dx,$$ $$ = {I_3} = \int\limits_0^1 {{2^{{x^2}}}} dx,\,$$ $${I_4} = \int\limits_0^1 {{2^{{x^3}}}} dx\,\,$$ $$\forall 0 < x < 1,\,{x^2} > {x^3}$$ $$ \Rightarrow \int\limits_0^1 {{2^{{x^2}}}} \,dx > \int\limits_0^1 {{2^{{x^3}}}} dx$$ and $$\int\limits_1^2 {{2^{{x^3}}}dx} > \int\limits_1^2 {{2^{{x^2}}}dx} $$ $$ \Rightarrow {I_1} > {I_2}$$ and ${I_4} > {I_3}$

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