Why is there always constant of integration when you evaluate an indefinite integral?

When you are asked to integrate a function f(x), you are really being asked the question: "what function F(x) exists such that when you take its derivative, you are left with f(x)?"

Let us first consider differentiation.

Let F(x)=x²

We know the derivative of this is f(x)=2x but what if F(x)=x²+5? 

It turns out the derivative of this is also f(x)=2x. That is because the derivative of 5 is 0 and so that disappears from the derivative.

In fact the derivative of any constant is 0 so the derivative of F(x)=x²+C (where C is any real number) is f(x)=2x

So now let us talk about integration.

We know that integrating f(x)=2x gives F(x)=x² because when you differentiate F(x) you are left with f(x). But this is also true for F(x)=x²+5  or in fact for F(x)=x²+C (where C is any real number).

And therefore the fact that the derivative of any constant is 0 is the origin of the constant of integration.