Use the substitution u = 2^x to find the exact value of ⌠(2^x)/(2^x +1)^2 dx between 1 and 0.

First thing to notice here is that this question tells us exactly what method they want us to use, subsitution, and what to substitute. Hence the first step is to turn the integral provided which is currently in terms of x into one in terms of u.
Using the information provided we can re-write u = 2^x = e^(2lnx). This makes it easier to differentate into du/dx = (2^x)ln2. We can now use this to substitute dx in terms of u into the integral and replacing all 2^x 's with u's. After some cancellation it equals 1/ln2 ⌠1/(u+1)^2 du. Note how I've kept the 1/ln2 out of the integral because it is a constant. This is a good habit to keep in general as it ensures that we do not do any more work than we need to later in the question. After some simple integration we are left with the equation -1/(ln2)(u+1) + C where C is a constant. (Since this question asks about definite integration this C will disappear later once boundary conditions are subbed in)
From here, we need to sub in our boundary conditions. But, CAUTION, our boundary conditions x = 1 and x = 0 are in terms of x, so we need to change them to u using our original equation u = 2^x. Hence our new conditions are u = 1 and u = 2. Substiting this into our new equation in terms of u gives us the exact answer 1/6ln2. Alternatively you can also revert our equation back in terms of x and not change out boundary conditions in terms of u. Both give the same answer.