Integrals "substitution approach"

Set u:

u = x^2 - 2

Differentiate both sides:

du = 2x*dx

Solve for dx:

du/(2x) = dx

Replace x^2 - 2 with u and dx with du/(2x):

INT(x*SQRT(x^2-2)dx,2,3) = INT(x*SQRT(u)du/(2x),?,?)

Then all your x's go away (which is necessary):

INT(1/2*SQRT(u)du, ?, ?)

Solve for your new bounds:

UB = 3^2 - 2 = 7, LB = 2^2 - 2 = 2

INT(1/2*SQRT(u)du, 2, 7)

Then you just have a simple polynomial integral (add 1 to u's exponent, divide by new exponent):

INT(1/2*u^(1/2)du, 2, 7) = 1/2*u^(3/2)/(3/2) | [2,7]
1/3*u^(3/2) | [2,7] = 1/3( 7^(3/2) - 2^(3/2) ) = 5.23

Have fun.