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.
