Answer
$$f\left( x \right) = \frac{{3{x^2}}}{2} - \frac{{\cos \pi x}}{\pi } + \frac{{1 - 3\pi }}{\pi }$$
Work Step by Step
$$\eqalign{
& f'\left( x \right) = 3x + \sin \pi x;{\text{ }}f\left( 2 \right) = 3 \cr
& {\text{Calculating the general solution}} \cr
& f\left( x \right) = \int {f'\left( x \right)} dx \cr
& f\left( x \right) = \int {\left( {3x + \sin \pi x} \right)} dx \cr
& f\left( x \right) = \frac{{3{x^2}}}{2} - \frac{1}{\pi }\cos \pi x + C \cr
& {\text{Calculating the particular solution for }}f\left( 2 \right) = 3 \cr
& 3 = \frac{{3{{\left( 2 \right)}^2}}}{2} - \frac{1}{\pi }\cos \left( {2\pi } \right) + C \cr
& 3= 6 - \frac{1}{\pi } + C \cr
& - 3+ \frac{1}{\pi } = C \cr
& {\text{The particular solution is}} \cr
& f\left( x \right) = \frac{{3{x^2}}}{2} - \frac{1}{\pi }\cos \pi x - 3+ \frac{1}{\pi } \cr
& f\left( x \right) = \frac{{3{x^2}}}{2} - \frac{{\cos \pi x}}{\pi } + \frac{{1 - 3\pi }}{\pi } \cr
& {\text{Graphing general solutions for }}C = - 4,{\text{ 1, 1 and the particular}} \cr
& {\text{solution }}f\left( x \right) = \frac{{3{x^2}}}{2} - \frac{{\cos \pi x}}{\pi } + \frac{{1 - 3\pi }}{\pi } \cr} $$
