Constant after integration

Question

What is the true significance of the constant "c" that we add after we integrate a curve without applying limits?

Answer 1

Integration is reverse process of Differentiation. Suppose, f(X) = 2X+ 3

f'(X) = 2 ---->(1)

Again, f(X) = 2X , f'(X) = 2---->(2)

Now, if you integrate 2 , we take integral value as

2x+c (because of What I mentioned above. Since, derivative of constant term is 0)

Hence, we don't know wheather it has constant term or not ! That's why we take a arbitrary constant C.

Answer 2

Actually, talking in $\Bbb R^2$ after integrating when we obtain a function $g(x)+c$ the set I={$g(x)+c | \forall c\in \Bbb R $} it indicates the entire Family of Curves . But when we fix the quantity 'c' to a fixed real number , we are actually indicating a specific curve out of the Family of Curves .That's the role of the integrating constant.