Don't worry lol, PF is here to teach =]
Ok first of all, substitution is basically just replacing a term with nicer looking terms, so that its easier to see how to continue. ( x+ 2\pi + xy)^5 + 2( x+ 2\pi + xy)^4 - 6( x+ 2\pi + xy) might seem a bit daunting, but looks much nicer when its u^5 + 2u^4 -6u, with the obvious substitution. It's sort of the same for integration.
The shorthand
\int^b_a f(x+c) dx really means \int^{x=b}_{x=a} f(x+c) dx.
So when we make the substitution u= x+c, we have to make sure every piece of information becomes in terms of u. u^5+ 2u^4 -6( x+ 2\pi + xy) is quite useless, its only when all parts are in terms of u are things easier.
So first we take care of the bounds. x=b, so u=b+c. x=a, so u=a+c. f(x+c) = f(u). du=dx.
Replacing all the parts of the integral with those, it becomes \int^{u= b+c}_{u=a+c} f(u) du, which in shorter notation, becomes \int^{b+c}_{a+c} f(u) du
All this without finding anti-derivatives. You should probably know that most of the theory of Integration can be done without finding anti-derivatives, its just a helpful tool that we can evaluate them with.
