The second uses substitution to convert the entire integral into one in terms of a new variable, u, as follows:
There is no set rule for how to make this choice, so you may have to make several tries. A good rule of thumb is that if there is a function of something other than x, (e.g., cos(2x+1)) or a power of some function of x or a root of a function of x, then these functions are good candidates for u.
This expression should appear as part of the integral. If it doesn't, you may have chosen a bad u. If the integral is missing some constant multiplier, you can fix it by multiplying by this factor inside the integral and dividing by this factor outside the integral.
That is, compute u when
x = aand when x = b. These will become the new limits of integration.
You should be left with no x's in the integral, and the limits of integration will have changed.
You never go back to x's using this method. Once you have converted the limits, there is no need to go back to the x's.
Note: The second method is the one I recommend, as it is usually shorter and easier to handle, provided you remember to adjust the limits of integration. If you don't, however, you will run into serious trouble! If this happens to you a lot, you may prefer the first method.