Find a function f (x) for which the function F is one of the antiderivatives on R if cos (pi / 3-2x) -arctg x + 2.

Since finding the antiderivative is a process inverse to differentiation, then in order to find a function for which this one will be one of the antiderivatives, we find the derivative of this function:

F (x) = cos (pi / 3 – 2x) – arctgx + 2;

F ‘(x) = (cos (pi / 3 – 2x) – arctgx + 2)’ = (cos (pi / 3 – 2x)) ‘- (arctgx)’ + 2 ‘= -sin (pi / 3 – 2x ) * (-2) – 1 / (1 + x ^ 2) = 2sin (pi / 3 – 2x) – 1 / (1 + x ^ 2).