Suppose that f(1)=2, f(4)=7, f'(1)=5, f'(4)=3, and f'' is continuous. Find the value of $\displaystyle \int_{1}^{4} f''(x) \,dx$
If you integrate f''(x) with respect to x you obviously get f'(x) (plus a constant which I will ignore since you have a definite integral). Therefore the given definite integral is equal to f'(4) - f'(1) = ....