Proving the second fundamental theorem of calculus?

[Vitani11]

Homework Statement

Show that D_x∫f(u)du = f(x) Where the integral is evaluated from a to x. (Hint: Do Taylor expansion of f(u) around x).

Homework Equations

None

The Attempt at a Solution

I have

... = D_x(F(u)+C) = D_x(F(x-a)+C) = d_xF(x) - d_xF(a) = f(x)-f(a). My problem is that it should be only f(x), not f(x) - f(a). I did a taylor expansion of f(u) around x and I'm not sure how that is supposed to help me...

 

[Vitani11]

I know that for an integral F(a) (where a is not a variable) the derivative of the integral would be 0, and that's why it would not be included in the final answer, but I don't know how to show that.

 

[haruspex]

Dx(F(u)+C)

You are starting with $\int_a^xf(u)du$, so that is a function of x (or maybe of a and x). It is not a function of u.