Answer
$$R\left( x \right) = 5.3\sin x + 1.71$$
Work Step by Step
$$\eqalign{
& r\left( x \right) = 5.3\cos \left( x \right);\,\,\,\,R\left( {3.14} \right) = 2 \cr
& \cr
& {\text{Write a formula }}R\left( x \right){\text{ for the antiderivative of }}r\left( x \right) \cr
& R\left( x \right) = \int {5.3\cos \left( x \right)} dx \cr
& R\left( x \right) = 5.3\int {\cos \left( x \right)} dx \cr
& \cr
& {\text{integrate }} \cr
& R\left( x \right) = 5.3\sin x + C \cr
& \cr
& {\text{Use the condition }}R\left( {3.14} \right) = 2{\text{ to find }}C \cr
& 2 = 5.3\sin \left( {3.14} \right) + C \cr
& 2 = 0.29 + C \cr
& C = 1.71 \cr
& \cr
& {\text{The specific antiderivative of }}f{\text{ is}} \cr
& R\left( x \right) = 5.3\sin x + 1.71 \cr} $$
