Riemann Integrale function?

Question

Is there any Riemann integrable bounded function on a closed and bounded interval [a,b] which has primitive function and it does not satisfy the Second fundamental theorem of calculus?

Sorry, I made a mistake in the title.
The correct title is: Riemann Integrable function?

Answer 1

No. Check out the corollary of the second fundamental theorem in:
http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus

It applies because of your constraints (function is bounded and Riemann integrable over a closed and bounded interval).