In integration, what does the +c mean and why does it disappear if you have limits?

When you differentiate, the constant disappears, because it is not dependent of the variable. So when you integrate, you have to add the constant again. However, you can't know the vaulue of c without being further information.
When you integrate with limits, for example x^2+x with the limits of 1 and 4. Integration: 1/3x^3+0,5x^2 + c. 
with limits: 1/3(4)^3 + 0,5(4)^2 + c - 1/3(1)^3 + 0,5(1)^2 + c --> c-c --> the c disappears.