TITLE: Why is there a constant of integration?
QUESTION [7 upvotes]: $\int (2x^2 +1)dx = \dfrac{2}{3}x^3 + x +C$.
This is a simple integral, and as you all know, the answer to an integral will always have a '$+C$' at the end, the constant of integration.
Why is it there?

REPLY [2 votes]: There are some extremely complicated answers here that are probably going to confuse the person asking the question (since it appears they have only just learned what integration is!).
Basically you were probably told by your teacher or by a book that differentiation and integration are inverses to each other. That is not quite true. When differentiating you lose information on what the original function was.
e.g. if we differentiate $x^2$ and $x^2 + 49730871208730750$ we get the same thing. This is due to the fact that the graphs "look the same" but one is a vertically shifted version of the other.
The derivative tells you the gradient of a curve at various points where it is defined. Intuitively you can see how this explains the behaviour above.
When integrating we are trying to "undo" the derivative to get one single function back...but which vertical shifting should we choose?
The answer is: It is impossible to know which one you started with. The best we can do is to say the answer is something with an ambiguity for the information that was lost, i.e. upto a constant $C$.