Question

1. 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:

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