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)