When integrating, why do we add a constant to the resulting equation?

The +c is to represent the loss in information after differentiation. Remember, integration is just the reverse of differentiation. Say we had this function:

f(x) = 2x^2 + 1 And we differentiate: f'(x) = 4x

Now take this second function: g(x) = 2x^2 + 4 And differentiating gives us: g'(x) = 4x

We can see that g'(x) = f'(x). So, if we try and integrate 4x, what do we get? Would it be 2x^2 + 1, or 2x^2 + 4?

The answer is it could be either. Or 2x^2 + 3. Or 2x^2 + 109823.1203981! There are infinite solutions to integration, depending on how you got there from differentiating. That's why we add the +c - to represent all the different possibilities.