Your work is correct but there're other faster ways, but since you told us you don't need additional hints, then I'll stay away.
The definite integral from 1 to 10 of x/x^2
and the indefinite integral of x+2/the square root of x^2+4x
For the first one I THINK u would be x^2 and thus du would be 2x and I'd have to divide it by two so then I'm left with 1/2 the integral of...what? Would it just be 1/u?
And for the second again I think I know to use x^2+4x as u making du 2x+4 so I'd have 1/2 the integral of 1/the square root of u?
If I'm correct so far just let me know and I'll keep working, preferably don't give me any other hints, but if I am wrong so far please do tell me where I'm going wrong!