Why a constant is written after integrating?

Answer 1

Integration is the opposite of differentiation (taking the derivative). The derivative of a constant is zero. Integration is also called antidifferentiation since integration and differentiation are opposites of each other. The derivative of x^2 is 2x. The antiderivative (integral) of 2x is x^2. However, the derivative of x^2 + 7 is also 2x.

Therefore, the antiderivative of 2x is x^2 + C, in general, where the constant C has to be determined from the context of the problem. In the above case, the constant happens to be C=7. We use integration to solve first order differential equations. When solving first order differential equations, like in "word problems", you must determine the integration constant using the initial conditions (ie the conditions we know to be true at t=0 - we usually know what these are), or the boundary conditions (ie the conditions we know to be true at x=0 and y=0).