Question

If $I_1={\int }_0^1{2}^{x^2}dx$, $I_2={\int }_0^1{2}^{x^3}dx$, $I_3={\int }_1^2{2}^{x^2}dx$ and $I_4={\int }_1^4{2}^{x^3}dx$

then which of the following is true?

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

Answer 1

The correct option is A $I_1>I_2$

For $0<x<1$, $x^2>x^3$ $\therefore$ $2^{x^2}>2^{x^3}$ and for $1<x<2$, $x^3>x^2$ $\therefore$ $2^{x^3}>2^{x^2}$

i.e. $2^{x^2}<2^{x^3}$ $\Rightarrow$ $I_3<I_4$

As $2^{x^2}>2^{x^3}$ $\therefore$ ${\int }_0^1{2}^{x^2}dx>{\int }_0^1{2}^{x^3}dx$

$\therefore$ $I_1>I_2$