Question

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
(A) ${I_3} > {I_4}$
(B) ${I_3} = {I_4}$
(C) ${I_1} > {I_2}$

Answer 1

Firstly, we will check that for the integrals ${I_1}$ and ${I_2}$, which function gives larger value in the interval $[0,1]$. Then we will check the same for the integrals ${I_3}$ and ${I_4}$. And on comparing, we will get the result.

Complete step by step solution:
First, we will take the interval ${I_1}$ and ${I_2}$ because they have the same interval 0 to 1.
Square of any value from the interval 0 to 1 is greater than the cube of that value.
$\Rightarrow {x^2} > {x^3}$ in the interval $[0,1]$.
Also, we can see that
i.e. $\int {{x^2}} > \int {{x^3}}$ in the interval $[0,1]$.
We can also see that
$\Rightarrow {2^{{x^2}}} > {2^{{x^3}}}$ in the interval $[0,1]$.
Therefore, the value of $\int\limits_0^1 {{2^{{x^2}}}dx}$ is greater than the value of $\int\limits_0^1 {{2^{{x^3}}}} dx$.
$\Rightarrow \int\limits_0^1 {{2^{{x^2}}}dx} > \int\limits_0^1 {{2^{{x^3}}}} dx$
Hence, ${I_1} > {I_2}$.
Now, we can see that the cube of any value in the interval $[1,2]$ is greater than the square of that value.
The value of ${x^3}$ in the interval $[1,2]$ is larger than the value of ${x^2}$.
Hence, ${x^3} > {x^2}$ in the interval $[1,2]$.
The integration of ${x^3}$ is greater than the integration of ${x^2}$ in the interval $[1,2]$.
Hence,
$\Rightarrow \int {{x^3}} > \int {{x^2}}$ in the interval $[1,2]$.
Therefore, the value of $\int\limits_1^2 {{2^{{x^3}}}dx}$ is greater than the value of $\int\limits_1^2 {{2^{{x^2}}}} dx$.
$\Rightarrow \int\limits_1^2 {{2^{{x^3}}}dx} > \int\limits_1^2 {{2^{{x^2}}}} dx$
Hence, ${I_4} > {I_3}$.
Therefore ${I_3} \ne {I_4}$

**Hence option (C) is correct
i.e. ${I_1} > {I_2}$ **

Note:
The given integrals are definite integrals. Definite integrals are the integrals whose limits are given. The definite integrals are mainly used to calculate the area of the region.