Suppose I have this function where a = 0 and b = .5 not sure on the latex tags for that
and I let
and
this is the part where I get confused...
Do I
or solve for dx completely
If I do it at the step above and continue on to
and plug in the corresponding values my answers do not come out correctly
Please type the question exactly as it appears in your textbook. Because it looks like you've omitted important imformation.
Memo to all members: It helps if you type all of the problem rather than just the bits of it you think are important. (The fact that important information is either overlooked or considered unimportant is often a major reason why people can't do the question they post). This saves time for everyone.
You aren't going to find an elementary antiderivative for that function... so unless you use estimation techniques or just put in into a calculator for the values of b=x that you want, I don't think you're going to get an answer
You could use taylor series to get as good an estimation as you'd like
As has been said - twice - the integral cannot be evaluated in closed form using elementary functions. So either important information from the problem statement is missing or you're expected to use technology to get approximate values for the integral (in which case the question will have said so). The fact that you might be expected to apply the Second FTC only lends weight to the fact that something is missing from the question as posted.
Perhaps the following link will put in context the many attempts that have been made to tell you that the integral cannot be done in closed form using elementary functions: Wolfram|Alpha
You will have to use technology to evaluate I(x) for x = 0.5, 1, 1.5, 2. Was that made clear in the question?
Except for when x = 0 in which case obviously I(0) = 0.