A function is analytic at a point c means the function have a convergent power series at point c and the function is infinitely differentiable at point c. For instance, if power series of the function converges to a constant. There is an example f(x)=|x|. It is not differentiable at x=0. It cannot even have MacLaurin series, which does not have a convergent power series at x=0.

It is useful, because it indicate us if a function-related question is able to solve by using the power series of it. As well, if a function is analytic at a point c, it is infinitely differentiable at point c; however, if a function is infinitely differentiable at point c, it may not be analytic at point c.

I think the part (f) in the first question is the most difficult one. Firstly it is hard to start since a part of the question looks like the derivative of arctan or arcsin, which is confusing. Actually the technique of integral by substitution should be used for the first step. After t is substituted by tan(x), the result is still complex, but it seems can be solved immediately. A while later, it is realized maybe another integral by substitution should be done, but it need to be revised first. Sec(x) and tan(x) should be written by 1/cos(x) and sin(x)/cos(x). Simplify the equation, finally we have a simple equation which is able to apply integral by substitution easily.

Tips:
1. For the first question which is basic stuff, it is definitely a good way to try three techniques for solving integrals.
2. Also, because of basic question, calm down and be patient. Trust yourself you can do it after a lot of practices. Of course, do practices.
3. Review the basic stuff like integral techniques first. It is the core techniques for the whole course.
4. Don’t forget the specific parts, such as absolute value and the thing like dt, dx or dh.