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

then which of the following is correct?

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

Answer 1

The correct option is A $I_1>I_2$

To find comparison between $I_1$ & $I_2$ and $I_3$ & $I_4$

We will have to draw the graph of $f\left(x\right)=2^{x^2}$ & $g\left(x\right)=2^{x^3}$ and study them for $0<x<1$ & $x>1$

Hence we can see that $f\left(x\right)>g\left(x\right)$ for $0<x<1$ and arc under the graph of $f\left(x\right)$ is also greater than that of $g\left(x\right)$.

Hence correct answer is $I_1>I_2\&I_3<I_4$