Question

The value of the integral ${\int }_1^3\sqrt{3+x^3}dx$ lies in the interval

• A. $(1,3)$
• B. $(2,30)$
• C. $(4,2\sqrt{30})$
• D. none of these

Answer 1

The correct option is C $(4,2\sqrt{30})$

As $f\left(x\right)=\sqrt{3+x^3}$ is monotonically increasing $\forall xϵ[1,3]$
For
$1<x<3\Rightarrow \sqrt{3+1^3}<\sqrt{3+x^3}<\sqrt{3+3^3}\Rightarrow 2<\sqrt{3+x^3}<\sqrt{30}\Rightarrow {\int }_1^{3}2dx\le {\int }_1^3\sqrt{3+x^3}dx\le {\int }_1^3\sqrt{30}dx\Rightarrow {2\left[x\right]}_1^3\le {\int }_1^3\sqrt{3+x^3}dx\le \sqrt{30}{\left[x\right]}_1^3\Rightarrow 4\le {\int }_1^3\sqrt{3+x^3}dx\le 2\sqrt{30}$