Answer
$$12-\frac{8\sqrt2}{9}$$
$$Or$$
$$10.743$$
Work Step by Step
$$\int_{1}^{2}2x^2\sqrt{x^3+1}dx$$
$u=x^3+1$
$du=3x^2dx$
$\frac{2}{3}du=2x^2dx$
$$\frac{2}{3} \int_{1}^{2} \sqrt u du$$
$\frac{2}{3} * _{1}^{2}|\frac{2}{3}u^{\frac{3}{2}}$
$\frac{4}{9} * _{1}^{2}|(x^3+1)^{\frac{3}{2}}$
$\frac{4}{9} (27-\sqrt8)$
$$=12-\frac{8\sqrt2}{9}=10.743$$
