y' = f'(x)
then y = f(x)
Let g(x) be any other integral of y' = f'(x)
That is, g'(x) = f'(x).
Now show that g(x) can differ from f(x) by at most a constant.
Then let w' = f'(x) - g'(x) = 0
Then w as a function of x must be w = constant
Hence g(x) = f(x) + constant.
Are they saying that because
w' = f'(x) - g'(x) = 0
w' = 0
Then because the slope of some curve is always zero, therefore it would be a line parrallel to the x-axis... that makes it a constant?
My real concern is.... how did they assume the g'(x) equals f'(x) ? Because if g(x) =/= f(x) then the difference of w' would be greater than a constant.....???