Answer
$$F\left( x \right) = - \cos x + 3x + 3 - 3\pi $$
Work Step by Step
$$\eqalign{
& F''\left( x \right) = \cos x,{\text{ }}F'\left( 0 \right) = 3,{\text{ }}F\left( \pi \right) = 4,{\text{ }} \cr
& F'\left( x \right) = \int {F''\left( x \right)} dx \cr
& {\text{Substitute }}F''\left( x \right) \cr
& F'\left( x \right) = \int {\cos xdx} \cr
& F'\left( x \right) = \sin x + C \cr
& {\text{Use the initial condition }}F'\left( 0 \right) = 3 \cr
& 3 = \sin \left( 0 \right) + C \cr
& C = 3,{\text{ then}} \cr
& F'\left( x \right) = \sin x + 3 \cr
& \cr
& F\left( x \right) = \int {F'\left( x \right)} dx \cr
& {\text{Substitute }}F'\left( x \right) \cr
& F\left( x \right) = \int {\left( {\sin x + 3} \right)dx} \cr
& F\left( x \right) = - \cos x + 3x + C \cr
& {\text{Use the initial condition }}F\left( \pi \right) = 4 \cr
& 4 = - \cos \left( \pi \right) + 3\left( \pi \right) + C \cr
& 4 = 1 + 3\pi + C \cr
& C = 3 - 3\pi ,{\text{ then}} \cr
& F\left( x \right) = - \cos x + 3x + 3 - 3\pi \cr} $$