How to integrate this

[Arrow_Raider]

Alright, I am not in calculus, but for some reason I am learning some calculus stuff on my own... I am actually currently a senior at highschool and taking a precalc math class.

When trying to integrate the problem located in Figure A at the bottom, I attempted integration by parts and came up with what is located in Figure B. However, when I have Mathematica calculate the integral, it gives me what is located in Figure C. When plotting these 2 functions(Fig B and Fig C), we get what is shown in Figure D. The function located in Figure D is has a restricted domain for any x values lower than 0. My function has the same domain restrictions as the original function, sqrt(x^2+x). As you can see, for the positive values of x, my function and the actual integral function do not match exactly. The error is greater than any possible calculation preciseness errors that could be caused when graphing. So my question is, what method would one use to go about integrating this function? Was using integration by parts the right idea but implemented wrong? Or is that not a valid method for this problem and a different method must be used?