Mathematics

Problem

A function

F

is defined by

F(x)=-\tfrac12\cos(2x)+3x+C,

where

C

is a real constant. Consider the integrand

g(x)=\sin(2x)+3

. (a) Verify that

F

is an antiderivative of

g

on the interval

[-\pi,\pi]

. (b) Determine the particular value of

C

for which

F(0)=1

. Present the resulting explicit formula for that particular antiderivative.