Meh, I couldn't get a picture. Let me try and explain it here.
First, you make u=x^2, that means that du=2xdx, therefore udu=2x^3dx, but you only need x^3dx in the numerator, so you divide everything by 2 and get udu/2 = x^3dx
so now you have the integral of udu/2(sqr(u+64)) which is the same as 1/2 times the integral of udu/sqr(u+64)
Now you make another substitution. Say that t=u+64, that means that dt=du and u=t-64. From here I'm gonna ignore the 1/2, I'll just solve the integral and multiply by 1/2 in the end. So making these changes you would get integral of (t-64)/sqr(t))dt.
If you separate that, it would be integral of t/sqr(t) dt - integral of 64/sqr(t)dt.
Solving those integrals (if you need to know how to solve them let me know, but they're rather simple) you would get 2/3t^(3/2)-64(2sqr(t) + c (never forget the constant!!)
and now multiplying by 1/2 that would be
1/3t^(3/2) - 64sqr(t) + 1/2c (this last thing is still a constant, let's name it k)
if you substitute back (remember t=u+64) you get
1/3(u+64)^3/2 - 64sqr(u+64) + k
and once again remember that u=x^2 so the answer is
1/3(x^2+64)^3/2 - 64sqr(x^2+64) + k
Let me know if you don't get something
Oh, sqr is square root.