Integrate : $I= \int \limits_1^3 \sqrt {3+x^3}.dx$

$\begin {array} {1 1} (a)\;I > 2 \sqrt {30} \\ (b)\;I < 2 \sqrt {30} \\ (c)\;I > 12 \\ (d)\;I < 12 \end {array}$

Its integral is not possible so find $min ^m \;\&\; max^m$ value :-
By maxima & minima method:-
$f(x) =\sqrt {3+x^3}$
$f'(x) =\large\frac{3x^2}{2 \sqrt {3+x^2}}$
$x= -1,3$
$f(1)= 2 m$
$f(3)=\sqrt {30} -M$
$2. (3-1) < \int\limits _1^3 \sqrt {3+x^3} dx < \int 30. (3-1)$
$4 < \int \limits_1^3 \sqrt {3+x^3} \;dx < 2 \sqrt {30}$
Hence b is the correct answer.