9. given the ( f^{prime}(x)=cos (x)+2 ) and ( f(pi)=5 ), find ( f(x) ).

Explanation:

Step1: Integrate (f'(x))

Integrate (\cos(x)+2) with respect to (x). Using the integral rules (\int\cos(x)dx=\sin(x)+C_1) and (\int 2dx = 2x + C_2), we get (f(x)=\sin(x)+2x + C) (where (C = C_1 + C_2)).

Step2: Use the initial condition (f(\pi)=5)

Substitute (x = \pi) into (f(x)=\sin(x)+2x + C). We have (f(\pi)=\sin(\pi)+2\pi + C). Since (\sin(\pi)=0), then (0 + 2\pi+C=5). Solve for (C): (C=5 - 2\pi).

Answer:

(f(x)=\sin(x)+2x+5 - 2\pi)