u=the square root of x+2

Then 

x=u squared minus 2

then

When you integrate the above, you're right with u=x+2.

(integrate) x/(sqrt(x+2)) dx

(integrate) x/sqrt(u) dx, where u=x+2 and du=dx.

So you'd get:

(integrate) x/sqrt(u) du

I think your question is what to do with the numerator.  Keep in mind that you set u=x+2.  Therefore, x must equal... u-2.  Subtract 2 from both sides.

(integrate) (u-2)/sqrt(u) du

Split the numerator to get:

(integrate) u/sqrt(u) - 2/sqrt(u)

(integrate) u^(1/2) - 2u^(-1/2)

Then you'd get 2/3(u)^(3/2) - 4u^(1/2) + C.

Substituting back, you'd get 2/3(x+2)^(3/2) - 4(x+2)^(1/2) + C.

Hopefully I've answered your question!  Feel free to ping me if you have any questions from my explanation.

 So when you determine what "u" should be you can also determine what "x" is as well and substitute that into the numerator in this type of problem? Thank you for the help.