Answer
$$f\left( x \right) = {x^3} - x + 2$$
Work Step by Step
$$\eqalign{
& f'\left( x \right) = 3{x^2} - 1;{\text{ }}f\left( 1 \right) = 2 \cr
& {\text{Calculating the general solution}} \cr
& f\left( x \right) = \int {f'\left( x \right)} dx \cr
& f\left( x \right) = \int {\left( {3{x^2} - 1} \right)} dx \cr
& f\left( x \right) = {x^3} - x + C \cr
& {\text{Calculating the particular solution for }}f\left( 1 \right) = 2 \cr
& 2 = {\left( 1 \right)^3} - \left( 1 \right) + C \cr
& 2 = C \cr
& {\text{The particular solution is}} \cr
& f\left( x \right) = {x^3} - x + 2 \cr
& \cr
& {\text{Graphing general solutions for }}C = 0,{\text{ 1, 4 and the particular}} \cr
& {\text{solution }}f\left( x \right) = {x^3} - x + 2 \cr} $$