Question

If $I={\int }_0^1\sqrt{1+x^3}dx$ then

• A. $I>2$
• B. $I\ne \frac{\sqrt{5}}{2}$
• C. $I>1$
• D. none of these

Answer 1

Answer: (C)

$I>1$

As $f\left(x\right)=\sqrt{1+x^3}$ monotonically increasing $\forall x\in [0,1]$
Now for $0<x<1$

$\Rightarrow \sqrt{1+0^3}<\sqrt{1+x^3}<\sqrt{1+1^3}$

$\Rightarrow 1<\sqrt{1+x^3}<\sqrt{2}$

$\Rightarrow {\int }_0^{1}1dx\le {\int }_0^1\sqrt{1+x^3}dx\le {\int }_0^1\sqrt{2}dx$

$\Rightarrow {\left[x\right]}_0^1\le {\int }_0^1\sqrt{1+x^3}dx\le \sqrt{2}{\left[x\right]}_0^1$

$\Rightarrow 1\le {\int }_0^1\sqrt{1+x^3}dx\le \sqrt{2}$