We have to find the value of the definite integral of x^2/sqrt (x^3 + 1) between the limits x = 2 and x = 3.

First we determine the indefinite integral and then substitute the values x = 3 and x = 2.

Int [ x^2 / sqrt (x^3 + 1) dx ]

let y = x^3 + 1

dy/3 = x^2*dx

=> Int[(1/3)*y^(-0.5) dy]

=> (2/3)*sqrt y + C

substitute y = x^3 + 1

=> (2/3)*sqrt (x^3 + 1) + C

Between the limits x = 2 and x = 3

(2/3)*sqrt (3^3 + 1) + C - (2/3)*sqrt (2^3 + 1) - C

=> (2/3)(sqrt 28 - sqrt 9)

=> (2/3)(2*sqrt 7 - 3)

The required value of the definite integral is (2/3)(2*sqrt 7 - 3)