Question

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

• A. $1<I<\sqrt{2}$
• B. $\frac{\sqrt{5}}{2}<I<\frac{\sqrt{7}}{2}$
• C. $\sqrt{2}<I<\frac{\sqrt{7}}{2}$
• D. $1<I<\frac{\sqrt{5}}{2}$

Answer 1

The correct option is A $1<I<\sqrt{2}$

Given:

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

Since, $0<x<1$

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

Integrating w.r.t $x$, we get

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

$\Rightarrow 1<I<\sqrt{2}$