Fundamental Theorem of Calculus states that the derivative of an INDEFINITE integral gives you the original function.
But this is a definite integral...

Be sure to multiply by the derivative of the bounds.

(2x)cos (e^x2) − (0)cos(e⁰) = 2xcose^x2

Refer to the hints in part (a), or... Notice that the definite integral equals a constant. What is the derivative of a constant?