Question

If $I_1={\int }_0^1{2}^{x^3}dx,I_2={\int }_0^1{2}^{x^2}dx,I_3={\int }_1^2{2}^{x^2}dx$ and $I_4={\int }_1^2{2}^{x^3}dx$, then

• A. $I_1>I_2$
• B. $I_2>I_1$
• C. $I_3>I_4$
• D. $I_1>I_3$

Answer 1

The correct option is A $I_1>I_2$

Given :-

$I_1={\int }_0^1{2}^{x^3}dx,I_2={\int }_0^1{2}^{x^3}dx,I_3={\int }_1^2{2}^{x^2}dx$ and $I_4={\int }_1^2{2}^{x^3}dx$,

As we know $x^2<x^3$ for $x\in (1,2)$ , So $2^{x^3}$ will also be grater than $2^{x^2}$ for $x\in (0,1)$ and $x\in (1,2)$

$\therefore I_1>I_2$

hence $I_3={\int }_1^2{2}^{x^2}dx$ < $I_4={\int }_1^2{2}^{x^3}dx$

And also $2^{x^2}$ when $x\in (1,2)$ is grater than $2^{x^3}$ for $x\in (0,1)$ hence $I_1={\int }_0^1{2}^{x^3}dx$ < $I_3={\int }_1^2{2}^{x^2}dx$

Hence, $I_1>I_2$ is correct.