Question

lf $I_1={\int }_0^1{2}^{x^2}$ 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

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

Answer 1

Answer: (B)

$I_1>I_2$

$I_1={\int }_0^1{2}^{x^2}dx,I_2={\int }_0^1{2}^{x^3}dx,$
$I_3={\int }_1^2{2}^{x^2}dx,I_4={\int }_1^2{2}^{x^3}dx.$
$\forall 0<x<1,x^2>x^3$
$\Rightarrow {\int }_1^2{2}^{x^2}dx>{\int }_1^2{2}^{x^3}dx.$$\Rightarrow I_1>I_2$

And for $1<x<2$ $x^2<x^3$

$\Rightarrow {\int }_1^2{2}^{x^2}<{\int }_1^2{2}^{x^3}\Rightarrow I_3<I_4$