When an function is analytic at c, it means that this function can be differentiable infinitely and it is converges to f(x) for x near c. This definition is useful because it can change a function into a infinite power series. This is more convenient for us to figure out many problems, such as whether this function converges or estimate decimal result.

A function differentiable infinitely does not mean it is analytic, such as problem.

However, an analytic function can be differentiable infinitely.

After I review my midterm exam, I found that I lose many points on the last step of integration. Every time when I finished the simplification part and I calculate the antiderivative of  the function I get a wrong answer for the miscalculation. For example, in the (f) question of first part, I simplified it into one over u to the power three and its antiderivative should be minus one over two times one over u to the power two but I got four instead, which leads to a wrong answer.

From the midterm, I noticed that this kind of mistakes are the main factor of losing mark. In order to solve this kind of problem, I should do more practice on antiderivative and develop a sense to those function, which can help me figure problems quickly during the final exam. In addition, I should get familiar with the antiderivative of the common functions, such as power function or trigonometric function.